Did you put your Retroarch folder in Program Files? That can cause write-protection problems like this (log.txt not being created). Running everything as an administrator can usually fix this, but it’s much better to just move your install to a normal folder (your user folder, for example). Also maybe try creating a blank log.txt file in your RA directory if that doesn’t work.
If you don’t have a bios file and need one, usually nothing works, but hey maybe. If you’re using mGBA (recommended), VBA Next, or gpSP you need gba_bios.bin in your system dir. It needs to be called “gba_bios.bin”, otherwise RA won’t be able to find it. By default, your system dir is simply retroarch/system/, but you can change it in options if you ever want to. This is also the way you would check where the directory is if you forget. (It’s also in retroarch.cfg.)
VBA-M and Meteor don’t need a bios file, so if you’re using either of those cores, it can’t possibly be the problem. In that case, maybe try another GBA emulator core? Don’t forget about the bios if you take that path.
For future reference, information about the bios etc. for each core can be found on the wiki.
Also, sometimes you need to explicitly choose a core, then open a file. If I have a rom saved as a .rom or a .bin, for example, there are multiple cores that can open it because of its generic extension (a .zip could be a NES rom, a SNES rom, an arcade game, etc.). So, I need to choose the correct core for the rom ahead of time. Otherwise, RA freaks out and crashes, since it can’t tell what core to use. You can avoid this with playlists, but that’s another story, involving romsets, CRC hashes, and scripts. There are other threads with more detail on that. (In other words: Load core > (…) > Load content > Select file > (…).)
One last thing: you said you’re using the latest build, by which I assume you mean a nightly. Using them is usually fine, but sometimes they contain weird bugs or are missing features, since they are being actively developed, so they aren’t guaranteed to be good to use. I usually stick with the stable builds, and try the nightlies only to see if bugs have been fixed sometimes. So maybe try out 1.3.6 stable with a blank config file (default settings) and see how that goes.
Welcome to Retroarch! I hope you like troubleshooting.