Government has mineral & property rights, i.e., gold & cannons, e.g., man digs up circa 1760’s cannon in side of back yard hill & Uncle Sam claims it.
No they don’t. Unless you’re in some other country. Stuff may fall under the antiquities act if it’s on federal land, but in the west, you can claim minerals on open federal land and they’re yours (pursuant to 1872 mining law et. al.).
East operates under leases.
Private property is also private mineral rights unless they’ve been separated in the past (See the many whiners in N and S Dakota with respect to neighbors making a mint on oil rights while they get jack because they bought land for less that didn’t come with them).