Pretty much all of them bar the supercard rumble are compatible with the GBA in the sense that if you can get it to fit in the slot it will boot and run whatever it can run.
Supercards (and their clones, team cyclops made a clone as their first flash cart and if you count it the very first GBA flash carts which were things from the GBC era slightly retooled) have a great many issues with a lot of games, every other GBA flash cart will run whatever it can fit give or take
http://gbatemp.net/threads/buying-a-gba-flash-cart-in-2013.341203/page-18#post-4756995 which is the same for all flash carts really and in all but two edge cases quite solvable.
Battery. All of them bar supercards which have their own very serious/annoying issues with saving use SRAM, this means they have a battery. Lifetime is in the order of several years, changing them bar one rare type of GBA flash cart involves soldering. Though it is the type of soldering that anybody that can claim to solder electronics can do.
These days you will be lucky to find GBA flash carts other than fire cards (don't, compatibility wise they beat supercards but they are very small and extremely fiddly to operate), supercards (don't), DS expansion packs like the EZ 3 in 1 (great if you have a DS, can't suggest it for use in a GBA though), possibly still an EZ4 lite deluxe here and there (will need to mod the case a bit) or the EZ4 (easier than a few months ago but still not trivial, will also want a miniSD card).
Similarly I do have to mention if you find a M3 professional/m3 pro or EZ4 lite compact they have limited GBA capabilities so avoid them.
Short version. You are looking for an EZ4 and a miniSD to go in it.