State requirements are centerfire-only, .22 caliber or larger, and 4" barrel or longer. Those are the legal requirements. Ethical requirements are often stricter, but not so easy to codify. Lesser cartridges/barrel lengths are often going to have shorter effective ranges and/or more critical requirements for shot placement.

Have you shot deer or other medium game with a 357? What is the basis of your considering it a bare minimum?

As a handgun hunter, what do you hunt with a handgun and what handgun do you hunt with?