Depending on function it serves,I'd call it a cam or bushing.

I would just replace it. You can still buy doorknobs like that. Its not really all that old. I have never seen parts for doorknobs, but, maybe, can find them online somewhere. However, the cost of the part, could be equal too the price of a new doorknob.

That's a very common part. Just buy an old lockset for cheap on eBay and substitute the innards.

Our local ReStore has parts like that: old disassembled latchsets.

Does anybody know the name of this particular interior door knob set?

Or perhaps name of a period?

Thank you!