Welcome!
"Proof" is a type of coin manufacture. Basically the dies (the metal outlines of the design used to strike the coins) are specially prepared, usually with mirror like fields (the flat bits of the design). Proof coins allow you to see the design in full detail. Sometimes proofs differ from circulation strikes. But sometimes it's difficult to tell a proof that's not cased from a good early strike from unworn dies.
"Mint" can be exchanged with "uncirculated" and just means that the coin hasn't knocked around. However coins aren't always well struck even straight off the production line. Some designs are generally shallow (the Mary Gillick bust on early Elizabeth II coins for example) and coins can pick up knocks ('bag marks') as they are packed for circulation. The Royal Mint specimen sets are 'mint'. But then you can find BU (Brilliant Uncirculated) coins in change if you're lucky. I got a 2013 £1 like that the other day.
As to mint and proof in circulation, well, there are certainly examples of proof coins that have been taken from sets and spent. Sometimes circulated examples are actually rarer than the unused coins! As far as I know the coins in ordinary sets are pretty much the same as circulating coins, just they've been boxed up before they've had a chance to see any wear.
That help?