1. I hear that there are custom firmwares and hacking methods that do not use flashcards at all. Do these methods allow you to play backups, or are they only good for homebrews?
Homebrew and backups
2. If these methods allow you to play backups, is there any difference in compatibility? I mean why would people even buy flashcards otherwise?
To be able to go online without installing bad tickets, and/or to not reach your home menu limit. Aside from that, since the softmods were released, people do not tend to buy flashcards.
3. Is there any advantage of using a flashcard over applying a non-flashcard hack?
Some believe that not having to modify your system (whether soft or not) is advantage enough. Some will say otherwise.
4. I believe that the hacks are available for certain official firmwares only (Below 9.0 or 9.2.0 or something). Is 2DS NWSMB2 edition hackable?
You didn't mention your system version, so how would I be able to answer that? Definitely yes with a hardmod, no idea about softmods until you give us more info.
5. Is there any significant drawback of using a 2ds compared to a 3ds (your personal opinion, battery, shape, lack of pro pad support, etc...)
In my opinion, the drawbacks are: small, shape, battery. Pro pad support only matters in a few games (Resident Evil, Monster Hunter, Majora's Mask, etc). I just don't like the 2DS.
6. If a non-flashcard hack is possible. Do I need any other hardware to perform the steps? How large are the backups? Should I get a new sd card with more
Again, you failed to let me know your system version. I refer you to the definitive (new)2/3DS/XL guide by Plailect: https://github.com/Plailect/Guide/wiki
Depending on your system software version, you may need an additional game to hack it (such as Cubic Ninja, OoT with save dongle, Smash Bros, etc), or you may need to hardmod your system to do a successful NAND dump and then inject an older native firm. It is all outlined in Plailect's guide.
A9LH + Luma3DS + LumaUpdater is the best way to go.