I'd like to install a slop sink in my basement. I can place it right next to the vertical main stack, and can connect the sink drain directly to it above the point where it enters the slab. I would also vent the drain line. Would this be allowed (in general) by most codes? Would a check valve between the trap and stack be required? I've never had any sewage backups, but then again I've never had any plumbing fixtures in the basement. My concern is being the lowest connection to the stack, waste and gas could back up through the slop sink drain.

May you connect your sink to the stack? Yes. May you use the stack as a wet vent for your sink? No.

My concern is being the lowest connection to the stack, waste and gas could back up through the slop sink drain. This is why we install "P" traps. Nothing could enter the sink unless you have a clog downstream and the it would back up out of the floor drain.

Would a check valve between the trap and stack be required? I've No,

I would also vent the drain line.
If you mean you're going to revent back into the stack for a vent , that's not acceptable. The sinks vent will have to revent back into a dry vent at least 6inches over the flood rim of the fixture that you're reventing back to.
Regards, Tom

No, I'd dry vent it by tying in to a vertical dry vent upstairs.