RR = Ranked Rounds - Any stroke play game that counts towards your average. These can be solo ranked rounds, WGT Tournaments (Brackets; Ready Gos; Weekly/Monthly WGT Stroke), or CC Tournaments. Only criteria is they can't be on Par3, Par 5, or Custom CC courses. Can't be solo games in Practice mode either, has to be Ranked. Getting into tournaments won't necessarily help you progress, unless you play the minimum number and achieve the minimum average score. If you are primarily coin player you may well have a better chance to advance tier by sticking with that.
The Rank you see Mondays is how many coins you've won or purchased vs. other players in your division. If you are Diamond I and finish in the top 10% for the week you will win some extra coins and be promoted to Diamond II, where the competition will likely be stiffer. If you click the Rank bar at the top of the Room Selection screen you can see your current status and then check periodically to see your progress, although it really means nothing in the grand scheme of things unless it's an ego thing for you.
As far as I know you can't turn on/off the gifting thing at the end of coin games. If you win, you may or may not receive the message asking to gift the loser a sleeve of balls. If you lose, the opponent may or may not receive a message asking if they want to gift you balls.
I used to think the type of ball that was involved in the gifting was dependent on the type of ball played during the match, but more and more lately I've been asked if I wanted to gift someone a sleeve of Titleists or Callaways when they had been playing a free starter ball during the match. Not sure the logic in that, other than a blatant money grab?