I'll try to cover each of the versions from start to finish. For this, I'm actually going to start with the Japanese release to give the full picture.
Kingdom Hearts (Japan)
Base game. Probably not exactly the one you know, though.
Kingdom Hearts (US & International)
Base game plus:
Additional Secret Bosses (Sephiroth, Kurt ...
Yes there are things that are missable, but nothing unique. There are no unique inventory items that are truly missable. While there was only one missable red trinity mark in Oogie's Mansion in the original KH, in Final Mix and consequently 1.5 HD Remix, that mark was moved to not be missable. As such, there are technically no missable items or trinity marks ...
Not really. Final Mix has new items, enemies, abilities etc. It is a much more better version than the original, although the original is not bad either. This mix is more of an upgrade version of normal.
Actually, at least in my case, KINGDOM HEARTS 358/2 Days showed on my games played list from just putting the disk in and going to the main screen (WITHOUT having launched any game from the main menu). It seems like that game shows on your games played list no matter what and the other two games will only show if you choose them from the main menu.
There aren't any functional differences in the games in terms of content. You're paying for the same content, just on the same disc versus two different discs. However, Square Enix significantly decreased load times for all of the games in the collection. Further, they play at 60 frames per second.
Trophies from the PS3 version don't carry over to the PS4 ...
You should be more than fine at that level for a decently normal difficulty fight with the right strategy. The key is to stay behind him and hit him on the back of his head. That will keep you out of the way of a lot of his attacks (though clearly not all). From there, it is just paying attention to when he starts the animations for the other moves.
So one thing to note is that the PS3 version KH 1.5 HD Remix does not actually come with both versions of the game. It only includes the Final Mix version (plus some additional updates; see this answer).
Included on the disc are 2 different games, plus cut-scenes from another. The two games are Kingdom Hearts Final Mix HD, and Kingdom Hearts: Chain of ...
You will be getting an EXP boost of plus 50%. I used to wonder the same thing so I equipped the items, killed an enemy worth a decent amount of exp and wrote it down and then killed the same enemy and got that down too. It ends up to be a boost of 50%. A good place to test it is Hollow Bastion just make a run through the last room (where the key hole ...
As per gameFAQS:
Unlockable How to Unlock
Kingdom Hearts 358/2 Days Obtain 358/2 Days Master Trophy
Kingdom Hearts Final Mix Theme Complete Kingdom Hearts Final Mix
Kingdom Hearts Re: Chain of Memories Complete Kingdom Hearts Re:...
The Red Trinity that has been moved to the gate near to Oogie's Manor appears to be somewhat glitchy. According to this forum thread, some steps to try are:
You may have to beat the mini-boss in the area and/or leave the room and come back before it'll work.
In that thread, leaving and re-entering helped the poster be able to use that red trinity mark.