You can. I'm not entirely sure it will work. The chainline won't be that good with such a long spindle.
You'll have to keep the old rear der to manage the slack between the two chainring sizes. A chain tensioner won't have enough slack to allow you to move between the two rings.
And you're going to want something up front to keep the chain on as well. Probably easiest to just keep the front der up there as a retention device.
An 8spd chain will work fine for this proposed drivetrain.
You didn't ask, but I don't think this is a great idea. Why not build it up as a single speed? Or a 1x8 or something? Having gears in the back is more useful and easier to build. What you propose is unwieldly and ultimately not going to serve you very well.