With only a few exceptions, every DS game is compatible with every other DS model, except 3DS games can only be played on a 3DS.
The DSi is no different from the regular model except it has no GBA slot-2, meaning any DS game that needs it (Guitar Hero) cannot be played on it.
No, you won't be able to do that unless the game/DSi you trade in are very recent games or limited edition/new.
No. Professor Layton and the Mask of Miracle is a game for the Nintendo 3DS and cannot be played on any version of the DS or DSi.
'Foto Showdown' is a 'DSi game', as it requires the use of the DSi camera. That means it would not be able to play on a regular DS or a lite, though technically it should work on a 3DS. As well as those, there's also DSiWare, which needs access to the DSi store, so again those are DSi specific and can't be played on a regular DS. Apart from those though, other games are either 'DS games' that work on every model, or DSi-enhanced games that work on every model, but have additional features on a DSi or 3DS.
There is no reason behind not being able to play Bolt on a Nintendo DSi unless the game isn't an offical licensed copy of the game since Bolt is a Nintendo DS game and therefore is capable of being played on all Nintendo DS systems which will include DSi, DSi XL, 3DS and 3DS XL.
Yes you can!
Well, any game that you can buy on the DSi store is not a DS game, such as "The Minis March Again." There is a DSi-only game which requires the camera to play, it is called Monster Finder and cannot be played on a regular DS (though it should work fine on a DSi).
Yes. It is a DS game, with DSi-enhancements. Because of this, the game is region-locked when played on a DSi, DSi XL or 3DS. This does not apply when playing on a DS or DS Lite.
Yes
No, 3DS games cannot be played on the Nintendo DS or DSi
All DS games will be able to be played on the DSi.
The DSi doesn't have a gameboy slot, so GB, GBC or GBA games can't be played on it at all.
Yes it can.