kingledion's answers are spot on, but I will add a little to that.
Lords, when in debt, and they frequently were, would sell everything they could to get out of it. They would tax in order to support their lifestyle, but they would also sell off rights to minerals, and it wasn't always to the King. Sometimes they sold them to a neighboring lord to get a cash infusion. Most often it was not the entire rights for all the property, but for a specific mine or area within their fiefdom.
The way it would work sometimes would be this. Lord A needs money now. Lord B has money and sees the potential of being able to sell the raw goods out of Lord A's mines. So he buys the rights to it, sometimes for a certain number of years. The miners, depending on the agreement, can be workers borrowed from Lord A (which Lord A can't afford to pay anyway) and Lord B would also agree to pay them for their labor, perhaps adding a supervisor of his own. Lord A might cut a deal with Lord B so that he can buy his own ore at a cut-rate price, up to a certain amount.
Now in an alternate situation, the King can see the same sort of opportunity and might also seize those rights if Lord A owes the crown a lot of money. Which, if Lord A hasn't been paying the taxes due to the crown because of his lavish lifestyle, he just might.
As to taxes, everything is taxed if it can be, including raw goods. The crown can choose to waive those taxes in favor of getting a certain amount of raw ore shipped to them for free or at a very reduced rate.
You also talked about the King's purview as far as coin minting is concerned. During Medieval times coins were actually not uniform, and many lords had their own coinage. England actually had one of the more uniform systems, early in the Medieval era making coinage a royal right. But even that didn't get a proper unification until 1279, with Edward I. The Florin became more of the standard than anything else because of the banking trade there.
In Europe proper, lords did indeed strike their own coin, as did abbeys and other places. In theory, the right to coin did reside with the King, but there were plenty of others granted that right by the King throughout the Middle ages. Here's a link with some of the basics of banking and coinage of the time.