If this doesn't have to be a PC, you might consider buying a hobby type single board computer that uses an Intel 8088 or 8086.
The monitor is driven by a graphics card. You can use an LCD monitor if it works with VGA output.
How the power supply works doesn't matter.
You should be able to by a pre-built hobbyist type single board computer.
You have to switch between banks of memory to access all of the video card's memory even when in graphics mode.
A monitor connects to a video card, not directly to a computer. You need a video card.
No, the program probably uses the direct x interface which is part of the video card driver that is installed into windows. The bios is not used.
It would be a lot easier to buy a hobby project board that just includes a processor and something like a small LCD display chip. I did a web search for "hobby single board project computer" and found a few hobby type boards. I would avoid the ones you have to wire up yourself.
If the goal is to learn pure assembly language, then why not start with some simple programs that run under a dos console window, using the standard c libraries to input from keyboard and output to video card? The point here is to learn assembly language programming in general, instead of learning to write device drivers.