Okay, turned out it was two separate issues.
Issue #1:
I installed a 64-bit version of Firefox and the ROBLOX launcher plugin (at least for Firefox) is 32-bit and can’t be used on the 64-bit program. I had actually run into this issue a while back when I tried using Waterfox (64-bit copy of Firefox), so when I noticed Firefox was 64-bit, the issue clicked.
Issue #2:
I don’t know? But logging into ROBLOX from the Edge browser, playing a game, and then opening studio fixed the problem. I’m now able to log into studio properly. Note: I did this before switching to 32-bit Firefox, so I don’t think issues 1 and 2 are related.