What wire should I use for a 100A subpanel to be located about 100 ft from house (200A) in a pole building. Is #3 copper (3 wires) and a #8 copper ground OK? Could I use 2-2-2-4 MHF?

The #3 copper is good for 100 amps but at 100' you're pushing it. Better to go with a #2 or even #1 with a #6 ground. I'm not sure what a pole building is, but you'll have to drive a ground rod at the subpanel. Is it a steel building?