Is there some way to access the 3D card with 16-bit ASM to use in QBasic, and would it be small enough to use in a module?
I think it's a 70/30 chance that it can't, and 99/1 chance that no one here knows how to do the asm for it
Nope, first of all you need to write a driver for every card major card that exists. And they're very different. Second, the registers on pci cards are memory mapped. And they're mapped much much higher then the 1 mb adress space. Infact if i remember correctly they're mapped in a virtual adress space above the 2 gigabyte barrier. So answer to your question is, impossible.
Yup. You have to do PMode to achieve that (32 bits programming), then interfacing with Glide is the best sollution for DOS. Voodoos implement it, and most old NVidia cards (with some drivers here and there) should...
But forget it... You'll need specific drivers. There is no such thing as VESA in the world of 3D cards.