My TI-Nspire CX has OS v. 4.5 on it. I installed Ndless and downloaded the gpSP-Nspire emulator on it along with the bios file needed. Yes, I put the bios in the same folder as the emulator.
There is an issue where my save states and .sav file reverted back to the beginning of the save (Pokemon Sapphire) for some reason, so I basically lost everything. How can I fix this?
Also, after the deletion of the save states mentioned above, when I tried loading emulator again, instead of showing the game selection screen like it always does, it showed a "missing bios" error message even though I had the bios in the folder, as mentioned above. Why would this have happened?

And... of course, the graphics = choppy/laggy. What settings would best work for my Nspire (off/automatic/manual, variation, etc.)?

Also, what other GBA emulators are there that I could replace gpSP-Nspire with, if I can't fix these problems?

This sounds like a question for calc84maniac, but he isn't around much anymore...
There are many things that can go wrong when running gpSP-Nspire... First of all, it's still a beta, so crashes and errors are to be expected. Secondly, some people have noticed that running it with any bios other than md5sum can cause issues, and finally, the ndless version might be wrong. I've personally had issues due to me using a newer OS with a newer version of Ndless, which is not what people were using back in 2012 when this was released. Of course, if it crashes for some reason, then the save file could be corrupted, and the emulator might decide to trash it. If this is the case, then I'm afraid there is nothing you can do to retrieve the lost progress. As for the performance, I've noticed this as well, and it is surprising when you compare the hardware of both a gameboy and the ti-nspire CX/CAS. Granted, emulating something takes a lot more resources than just running it on original hardware, but it is kind of strange to see it run considerably slower on newer hardware. Of course, off is the best setting to reduce lag, but honestly, the difference is marginal, so it won't really make a difference.
tl;dr: It is working as designed

I've got a problem with gpSP-Nspire where it works perfectly after resetting the calculator but then when I turn it off and then back on it tells me the BIOS is missing and I have to reset the TI-Nspire CX CAS again.

Firstly, welcome to Cemetech! Before you post to a topic, please check the date of the original topic first. This topic is almost three years old so you probably won't get any help here. This is also called necroposting and is looked down upon in most communities where threads don't get locked after some time.

Anyway, what you should do first, is to make sure that both ndless and gpsp are up to date. I see you already created an issue, but someone else had already created the same issue a few days ago. It would have been better if you had just subscribed to the old issue, because yours will be closed anyway, as it is a duplicate. Now all you can do is to wait for an answer.

You can also still try on the #ndless-dev IRC channel on EFnet, where you might get an answer faster.

EDIT: also, because the CX II has only recently started being supported, not all programs have been updated yet to support the new hardware. It's actually surprising that it's working this well.