1. There are no fancy models of DS lite. Go with whatever colour you like.
2. Not in any way that you care about, indeed not at all for the DS lite.
3. The DSi family does not have a GBA slot so won't be much good with your shiny new GBA cart. It is also marginally harder to source a DS slot flash cart or otherwise hack it -- the DSi and 3ds introduced checks for flash carts and not everybody kept up.
4. The DStwo is probably still king of the hill -- all the best features, all the best compatibility (not that there is a lot in it). They have not been made in a while so are harder to find, or otherwise more expensive than that chart there (the DStwo+ or the supercard 4 in 1 being the newer options here). I am not sure what flavour of R4 named device/clone we are suggesting these days. It was the R4i gold last I looked but it could be another now -- makers of them come and go, you also have "timebombs" to deal with in some of those. You could still get an R4 and run wood on it but you will be limited to 2 gigabyte microSD cards, and the AK2i is a nice thing for the DS lite.
Definitely don't get a gateway.
Hardware wise check for scratched screens, dodgy shoulder buttons, that volume control works, that the power button/slider still work (they are a bit weak on the DS lite), as it is easy you might as well check the cart slots don't have any bent pins (if they boot games you are good) and those are your main problems you encounter. There are others but they are quite rare.
Expanding on 3. The DS lite is probably one of the better ways to experience GBA games. The main downside is it has no link port if you want that sort of thing (link games, though you can dump your saves from a flash cart and use a PC emulator if you wanted, and things from the gamecube as well as a few oddities like using your GBA device to communicate with the PC/act as a controller). I could make a case for any of the other officially available common ways to play but few would argue against a DS lite for it.
