You're on the right track to use a diamond core bit for the stone. I don't know if you're planning to use a standard drill motor or if you're planning to use an angle grinder, but you're much better off using an angle grinder. The core bits will be longer and the grinder will give you more control versus a drill motor. I would also recommend to oversize your hole by a little bit and I'd drill a 5/8" hole instead.

For the sink mounting flange you'll almost certinly completely burn a bi-metal hole saw while attempting to drill through the stainless steel. You'd really want to use a carbide tipped hole saw to go through the stainless steel if you expect to use the hole saw more than once.