PGF's are gen 5 template gifts which are not Pokemon files.
Load them into your gen 5 save file with PokeGen, receive in game; export the Pokemon you receive -> load in PKHeX.
Those records are stored in the trainer data stats editable in the bottom right corner of the Trainer Info form.
As for which stat value is what, I don't know beyond the ones that currently have descriptions.
No, you're confusing DexNav count which is used in OR/AS. The chain counter is not in the save, it's in the RAM -- hence why when you reset your 3DS the chain doesn't continue.
You can edit the party; it's just that the injection method does not allow for it.
Battle Box is used for online play and official tournaments, hence why it is not editable.
Egg is not a species, it is a boolean state.
The only way to battle with eggs is via the Battle Box, which is not editable in PKHeX for obvious reasons. Feel free to move data to those slots in-game.
Manually toggling badges and unlocking locations on the map is a bad idea if the events that unlock them have not been completed.
You may click on badges to toggle them, but there has not been research done for location traveling.
Box Pokemon's stats are not stored -- they are generated when the data is displayed. Only Party Pokemon have battle stats, which is not editable by the RAM2sav injection exploit as Party Pokemon are loaded & operated elsewhere in memory.
1/29/15 - New Update:
- Changed: ramsav detection to allow users to store backups with unique names. Be sure it contains "ram" in the file name!
- Fixed: Gen4 Egg Locations should now load for pre-GenV origin mons. Thanks Omicron!
- Fixed: Unown forms M & F won't alter the gender label. Thanks Favna!
- Fixed: ram2sav usage with X/Y after OR/AS and not deleting the OR/AS save before dumping X/Y (data past sav length). Thanks trainboy2019!
- Added: Symbol selection for OT/Latest Handler like for Nicknames; access changed to control clicking the TextBox. Thanks ProfShroomish!
- Changed: FindIndex hunting to account for rare ramsav types. Thanks Crowhogan33 for reporting and SciresM for fixing!
Hold shift when exporting your ramsav; PKHeX will prompt for a target ramsav to export for that cart/system instead of for the the one that was loaded.