@No.one
1) As theSLAYER said, it's the official US Eon ticket e-card (included for event completion and as another option to get it).
2) e-berry where (apparently) removed from FRLG and Emerald. No cards where released to my knowledge and there's no space in the savegame to hold the berry data due to WonderCards being stored there. There's the slim chance that they moved it to another portion of the savegame, but without e-cards we can't know without reverse enginering the rom (which I'm not capable of).
3) This one is tricky. In RS Eon Ticket event used Mystery Event option to get it. In those games, a script is stored in the savegame AND the data for record mixing it is also stored in the save. When you talk with your dad at petalburg it gets executed, giving you the Eon ticket item and enabling the flags for the events, then the script gets erased from the save (so everything returns to normal).
Since emerald was released after RS and the Eon ticket for them was made, they put the Eon Ticket event script in-game. In japan, the oficial event was still carried out via Mystery Event, but instead of sending the script and record mixing data, it enabled the flag (only one!) in the save for the in-game event and stored the record mixing data.
In the official emerald event, you get handed out the ticket by the man in blue at the pokémon center (this is the in-game event).
The ticket events are scripted in a way that if one of the flags for the actual event or the item are already in the save they won't trigger again. This for example is what makes that if you hacked in (or used a glitch) to get the Eon Ticket item the event won't be enabled (if I recall properly).
Enabling the unofficial Eon Ticket event would yield the same results as mixing records because that button won't inject the data needed for sharing an object via mix records (you can check this yourself via extracting the mystery event from the save and opening it in the editor). There will be one difference: a flag that should only be possible to be enabled in the japanese version of the game will be enabled.
If you want an alternate method to get the eon ticket in emerald without mixing records that yields the same result, the easiest (and flashiest) way would be to get the rom script for the in-game eon ticket event and convert it into a wondercard, since wondercards can be tossed and shared, that would be the best way and would also feel like doing the event instead of just enabling the flags and giving the item via save editing.
4) For the TGC, we have the emerald counterpart and the WC text is different. For the Rocks america one we don't know if the text was different, it might have just been the standard "Mystic Ticket" we already have (the same card was distributed in Pokemon Center New York and Mexico, Rocks America might or might not have different text). The events themselves are the very same, so it doesn't really make a difference in-game. Would still be nice to have all the variants available though.