Welding table thickness

I *believe* hot vs cold won't make a big difference, though the cold working may contribute a little.

I think at 3/8, depending on how big it is, a lot of it will come down to the structure it's attached to.

Mine's about 2' x 4', 1/4" plate that wasn't entirely flat, but is much flatter now that it's welded to a 2" x 3" (IIRC) 0.125-wall apron that runs between the legs.

What you mean by "stay flat" is also a good question. What are you going to do to it that it needs to fight through to stay flat? Once 1/8" is flat, it'll stay pretty flat if you're not using it as a jig...

Note also the Certiflat type tables that are something like 3/16", but have a grid of webbing on the bottom that provides the rigidity. So, that's sort of the second data point for doing something other than relying on having a plate heavy enough to do all the work.