The R/B/Y re-releases are download only. The "sixth generation" games (X, Y, Omega Ruby, and Alpha Sapphire) can be obtained in both physical form (cartridges) and as downloads. The cost should be similar either way. In general, Nintendo games are tied to the system, not to your account, so if you lose your system, you can't get your games back unless you manage to transfer your Nintendo Network ID to a new system first (and if you want to upgrade your system, there's a fairly lengthy process to transfer everything). I don't know whether you've gotten consoles made by other companies recently, but Nintendo's account system is a lot less flexible, than, say, Steam, as far as I understand.
Regarding which system to get, it depends on what your price range is and what features you want. Personally, I tend to opt for the 3DS over the 2DS, but the 2DS is a cheaper alternative. For 3DS vs. 3DS XL, there is a considerable size difference. I have medium-sized hands, and I found the 3DS somewhat uncomfortable to hold, while I'm very happy with my XL. so if your hands aren't tiny, I would opt for an XL model. I don't know much about the New 3DS, as I don't own one and have never used one.
As for what games to get, if you want to relive your RBY days, those games were just re-released today for $10 apiece, and apparently will be able to transfer Pokemon to the upcoming seventh generation games, Sun and Moon. If you want one of the newer games, look at one of either Omega Ruby and Alpha Sapphire or X and Y. OR/AS have a few more features, so I'd probably recommend picking one of those, though X and Y have some nice features like trainer customization.
