theSLAYER

In Pokémon FireRed & LeafGreen,  a Magikarp can be purchased from a salesman in the Pokémon Center on Route 4, for 500 Pokédollars.

In Pokémon FireRed & LeafGreen, an Eevee can be obtained from the roof of the Celadon Mansion as a gift.

A modified emulator, that is able to connect a main game to a digital copy of the e-Reader, to send files to the intended main game.

Additionally, if you rather not use the .raws provided, we also have e-reader saves with the Eon Ticket already scanned into the e-reader.

Huge thanks to @javier_himura for mixing the VBA versions to craft this,
and also huge thanks to Team Fail, Hacky, Ajxpk, Gocario, Real.96, Suloku any other involved in Gen 3 Mystery Gift research thread, for the e-reader saves.

Original Thread and Instructions on Usage:

[We will not distribute any ROMs, and it is against our rules to ask for it]

In Pokémon Sapphire & Pokémon Emerald (when player selected Red at the TV at home when first entering Hall of Fame),
with the Eon Ticket (received from event in real life then from Norman in-game RS; received from Ruby & Sapphire via Record Mixing E),
players get to visit Southern Island and encounter the Legendary Latios.

Latios is the other Euo duo that wasn't roaming in the player's game, and it is holding the Soul Dew.

For players that wants to activate the event without Injecting the Mystery Event (EN/JP)RS/Gift Record Mixing (save)RS, read about the event flags here.

In Pokémon Ruby & Pokémon Emerald (when player selected Blue at the TV at home when first entering Hall of Fame),
with the Eon Ticket (received from event in real life then from Norman in-game RS; received from Ruby & Sapphire via Record Mixing E),
players get to visit Southern Island and encounter the Legendary Latias.

Latias is the other Euo duo that wasn't roaming in the player's game, and it is holding the Soul Dew.

For players that wants to activate the event without Injecting the Mystery Event (EN/JP)RS/Gift Record Mixing (save)RS, read about the event flags here.

Using the Aurora Ticket received from events on their copy of Pokémon FireRed & LeafGreen games,
players gain access to the Birth Island, and get a chance to capture the Mythical Deoxys!
Do note that the Aurora Ticket was never distributed to the Japanese copies of Pokémon Emerald Version.
For players who rather not use the Mystery Gift to activate the event, simply use PKHeX's Event Flags > Research, and
For Pokémon FireRed & LeafGreen: Check Flag 2122 and Flag 2123 , then proceed to add an Aurora Ticket into the Key Items of the Bag.
For Pokémon FireRed & LeafGreen, having the National Dex and Rainbow Pass may be required. More event flags here.

Using the Mystic Ticket received from events on their copy of Pokémon FireRed & LeafGreen, or Pokémon Emerald games,
players gain access to the Navel Rock, and get a chance to capture the Legendary Lugia!

For players who rather not use the Mystery Gift to activate the event, simply use PKHeX's Event Flags > Research, and
For Pokémon FireRed & LeafGreen: Check Flag 2122 and 2123
For Pokémon Emerald: Check Flag 0315 and 2272
then proceed to add a Mystic Ticket into the Key Items of the Bag.

For Pokémon FireRed & LeafGreen, having the National Dex and Rainbow Pass may be required. More event flags here.

Using the Mystic Ticket received from events on their copy of Pokémon FireRed & LeafGreen, or Pokémon Emerald games,
players gain access to the Navel Rock, and get a chance to capture the Legendary Ho-Oh!
For players who rather not use the Mystery Gift to activate the event, simply use PKHeX's Event Flags > Research, and
For Pokémon FireRed & LeafGreen: Check Flag 2122 and 2123
For Pokémon Emerald: Check Flag 0315 and 2272
then proceed to add a Mystic Ticket into the Key Items of the Bag.

For Pokémon FireRed & LeafGreen, having the National Dex and Rainbow Pass may be required. More event flags here.

In Pokémon Ruby & Pokémon Emerald (when player selected Blue at the TV at home when first entering Hall of Fame),
with the Eon Ticket (received from event in real life then from Norman in-game RS; received from Ruby & Sapphire via Record Mixing E),
players get to visit Southern Island and encounter the Legendary Latias.

Latias is the other Euo duo that wasn't roaming in the player's game, and it is holding the Soul Dew.

For players that wants to activate the event without Injecting the Mystery Event (EN/JP)RS/Gift Record Mixing (save)RS, read about the event flags here.

In Pokémon Sapphire & Pokémon Emerald (when player selected Red at the TV at home when first entering Hall of Fame),
with the Eon Ticket (received from event in real life then from Norman in-game RS; received from Ruby & Sapphire via Record Mixing E),
players get to visit Southern Island and encounter the Legendary Latios.

Latios is the other Euo duo that wasn't roaming in the player's game, and it is holding the Soul Dew.

For players that wants to activate the event without Injecting the Mystery Event (EN/JP)RS/Gift Record Mixing (save)RS, read about the event flags here.

The Old Sea Map was only available as an event Key Item exclusively for players with a Japanese copy of Pokémon Emerald,
which allowed players to reach the Faraway Island, and ultimately battle the Mythical Mew.

For players who rather not use the Mystery Gift to activate the event, simply use PKHeX's Event Flags > Research, and
For Pokémon Emerald: Check Flag 0316 and 2262, then proceed to add a Old Sea Map into the Key Items of the Bag.

This is the only Mew that can be encountered shiny legitimately. (excluding hacks or glitches)
ウルトラ Mew was a regular encounter, non-shiny.
ポケパ-ク Mew was RNG obtained by @suloku and @St. GIGA (and the OT was meant to mimic the PokePark event, and also Flawless IVs)
アイコ Mew was RNG obtained by @InsaneNutter
     Species   Mew   Nickname   (default, in )   OT   (recipient)   TID   (recipient)   Distribution   In-Game Encounter   Location   Faraway Island   Dates   (Various)   PID   Chance of Shiny   Games   Emerald                                                                Lv. 30    Nature   Random   Ability   Synchronize   Item   None    Moves       Pound   Transform   Mega Punch   Metronome
  Format Ver.1.0.6-2, Post Updated Date:20180512_0857

Using the Mystic Ticket received from events on their copy of Pokemon FireRed & LeafGreen, or Pokemon Emerald games,
players gain access to the Navel Rock, and get a chance to capture to Legendary Lugia!
For players who rather not use the Mystery Gift to activate the event, simply use PKHeX's Event Flags > Research, and
For Pokémon FireRed & LeafGreen: Check Flag 2122 and 2123
For Pokémon Emerald: Check Flag 0315 and 2272
then proceed to add a Mystic Ticket into the Key Items of the Bag.

The shiny Lugia here is obtained via Tool Assisted RNG.

Using the Mystic Ticket received from events on their copy of Pokemon FireRed & LeafGreen, or Pokemon Emerald games,
players gain access to the Navel Rock, and get a chance to capture to Legendary Ho-oh!
For players who rather not use the Mystery Gift to activate the event, simply use PKHeX's Event Flags > Research, and
For Pokémon FireRed & LeafGreen: Check Flag 2122 and 2123
For Pokémon Emerald: Check Flag 0315 and 2272
then proceed to add a Mystic Ticket into the Key Items of the Bag.

The shiny Ho-oh here is obtained via Tool Assisted RNG.

Original Creator
@suloku
 
XY friend safari usage
To be added
 
ORAS Mirage Island usage
This tool works by adding the wanted Mirage Spots to the PSS Passerby data.
Unlike standard daily mirage spots, players will not be able to access the hidden items.

However with this tool, players will be able to reach Crescent Isle much easily, to encounter Cresselia.

In order for this tool to work:
1. Make a fresh save for the day (Save in-game for that day, before you continue)
2. Use PKHeX to set the Player's Trainer ID to 00000,
3. Use this tool to set the ID of the islet you wanted.
4. Must remain as TID 00000 when soaring in the sky.
5. Land on the intend islet.
6. Save the game
7. You may now switch back to your previous TID
 
Original Research Thread
Research done covered Teleportation coordinates to the islands, and StreetPass Passerby Data for Mirage spots:

This gift was given away to participants of the 2019 International Challenge (February).
Participants were required to take part in at least 3 battles of the Online Competition, with a game synced game and account.
As this Online Competition was only available to Ultra Sun and Ultra Moon, the gift is also only receivable on Ultra Sun and Ultra Moon.
   Species   Tapu Bulu   Nickname   (default, save lang)   OT   Ula-Ula   TID   190222   Distribution   Serial Code   Location   the PGL   Dates   Mar 19 to Apr 30, 2019   PID   Shiny PID   Games   US & EU 3DS: USUM                                                                Lv. 60    Nature   Random   Ability   Grassy Surge (1)   Item   Grassy Seed   Bundled Item   None   Card Per Save   Only One Allowed    Moves   (ITALIC means Relearnable)   Nature's Madness   Zen Headbutt   Megahorn   Skull Bash   WC ID 1634: 「Tapu Bulu variocolor de regalo」's receiving text 0x38 This is a gift from the Pokémon Global Link! Please pick up your gift from the deliveryman in any Pokémon Center.

Classic Tapu Bulu variocolor de regalo Aquí tienes tu premio por haber participado
en el Torneo en Línea. ¡Disfruta de este
Tapu Bulu variocolor tan especial! Format Ver.1.0.7-0, Post Updated Date:20190320_0001

This gift was given away to participants of the 2019 International Challenge (February).
Participants were required to take part in at least 3 battles of the Online Competition, with a game synced game and account.
As this Online Competition was only available to Ultra Sun and Ultra Moon, the gift is also only receivable on Ultra Sun and Ultra Moon.
   Species   Tapu Bulu   Nickname   (default, save lang)   OT   Ula-Ula   TID   190222   Distribution   Serial Code   Location   the PGL   Dates   Mar 19 to Apr 30, 2019   PID   Shiny PID   Games   EU 3DS: USUM                                                                Lv. 60    Nature   Random   Ability   Grassy Surge (1)   Item   Grassy Seed   Bundled Item   None   Card Per Save   Only One Allowed    Moves   (ITALIC means Relearnable)   Nature's Madness   Zen Headbutt   Megahorn   Skull Bash   WC ID 1634: 「Un Tapu Bulu cromatico in regalo」's receiving text 0x38 This is a gift from the Pokémon Global Link! Please pick up your gift from the deliveryman in any Pokémon Center.

Classic Un Tapu Bulu cromatico in regalo Ecco la ricompensa per aver partecipato
alla Gara Online! Divertiti a lottare insieme
a questo rarissimo Tapu Bulu cromatico! Format Ver.1.0.7-0, Post Updated Date:20190320_0000

This gift was given away to participants of the 2019 International Challenge (February).
Participants were required to take part in at least 3 battles of the Online Competition, with a game synced game and account.
As this Online Competition was only available to Ultra Sun and Ultra Moon, the gift is also only receivable on Ultra Sun and Ultra Moon.
   Species   Tapu Bulu   Nickname   (default, save lang)   OT   Ula-Ula   TID   190222   Distribution   Serial Code   Location   the PGL   Dates   Mar 19 to Apr 30, 2019   PID   Shiny PID   Games   EU 3DS: USUM                                                                Lv. 60    Nature   Random   Ability   Grassy Surge (1)   Item   Grassy Seed   Bundled Item   None   Card Per Save   Only One Allowed    Moves   (ITALIC means Relearnable)   Nature's Madness   Zen Headbutt   Megahorn   Skull Bash   WC ID 1634: 「Schillerndes Kapu-Toro als Geschenk」's receiving text 0x38 This is a gift from the Pokémon Global Link! Please pick up your gift from the deliveryman in any Pokémon Center.

Classic Schillerndes Kapu-Toro als Geschenk Hier ist die Belohnung für deine Teilnahme am
letzten Online-Turnier. Viel Spaß mit diesem
seltenen Schillernden Kapu-Toro! Format Ver.1.0.7-0, Post Updated Date:20190319_2358

Spinda can normally be encountered at the Desert area of the Safari Zone, when 14 Plains-type block are placed,
such as Shrubbery, Red Flower and White Flower.
Using the Cute Charm Abuse (which requires player's to have a compatible TID/SID combo), it is possible to consistently encounter Shiny Spinda with the same spots.
For other Cute Charm'd Pokémon, visit this page.
Explanation:
 

Details below refer to the gift itself, not the file.
     Species   Rotom   Nickname   (default, in )   OT   (recipient)   TID   (recipient)   Distribution   In-Game Encounter   Location   Old Chateau   Dates   (Various)   PID   Chance of Shiny   Games   DP/Pt                                                                Lv. 20    Nature   Random   Ability   Levitate   Item   None    Moves       Thunder Shock   Confuse Ray   Uproar   Double Team
  Format Ver.1.0.6-2, Post Updated Date:20180424_2135

Besides events that distribute Phione (or a trade with My Pokémon Ranch),
Phione can only be obtained by breeding Ditto with Manaphy/Phione.

In Pokemon FireRed & LeafGreen, Moltres resides on the summit of Mt. Ember.

In Pokémon FireRed & LeafGreen, Zapdos can be encountered inside the abandoned Kanto Power Plant along Route 10.

In Pokémon FireRed & LeafGreen, Articuno can be encountered in the deep recesses of the Seafoam Islands.

This entry was transferred from from Pokémon Go to Let's Go Pikachu and Lets Go Eevee.
The Original Trainer for Pokémon Go was changed for privacy purposes.
This page also holds a Shiny Moltres that was captured by @wo0ts on September 9th 2018 during the three hour long Moltres Day event.
All gyms hosted Moltres as the raid boss, that was on for the full three hours.
After winning the raid there was a chance to encounter a shiny Moltres, which could then be transferred to Let's Go Pikachu and Lets Go Eevee.

This entry was transferred from from Pokémon Go to Let's Go Pikachu and Lets Go Eevee.
The Original Trainer for Pokémon Go was changed for privacy purposes.
This page also holds a Shiny Zapdos that was captured by @wo0ts on July 21st 2018 during the three hour long Zapdos Day event.
All gyms hosted Zapdos as the raid boss, that was on for the full three hours.
After winning the raid there was a chance to encounter a shiny Zapdos, which could then be transferred to Let's Go Pikachu and Lets Go Eevee.