Qualifying offer is more of a MLB concept than an OOTP one.
You can read about it at mlb.com, but the general idea is that instead of losing a free agent for nothing, you can make them a qualifying offer and then if you do still lose them you'll get a compensatory draft pick. I'm not a fan and don't play with it.
"Allow players to be purchased...", I think, means that a team in another league can pay you for your players instead of trading for them. I don't play with it either and wonder how it differs from the posting system like with Japan. My guess is the difference is with posting both the player and team get money and the player gets a new contract whereas with purchases the player is still under contract so just the team gets money.