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Yes, they are used all the time in the US.

The numbers on the bottom are the routing number and account number of the payer. Anyone can make and sell them, and anyone can buy them.

You have to order them. They don't just give them to you.

Discover gives free standard checks, as many as you want.

You have to pay for them but yes you can still get checks. My order of 80 checks should last me the rest of my life.

Absolutely. I currently rent from an elderly couple and the simplest method of paying rent is the classic check.