You can challenge or be challenged at any time (by friends or randoms). When you challenge someone, you and your squad (AI) plays against the other team (ALL AI, even if they are online).
From what I can see, it is all and no , as you can only level, and the stats do not seem to carry over to MP. As well, you get bonus XP time. When you play in Squads or MP you get a 10% bonus (e.g. 110 pts for a kill in TDM vs 100)
There is a Squad FAQ which may answer some of your other questions.