I think this could be a fun topic--especially for a relative beginner like me.  I play HU almost exclusively, so my answers describe HU play.

I don't always open 3BB, but I don't have very sophisticated thinking here.  Against a player that I perceive to be weak because he bets too aggressively (a lot of barreling and re-raising), I might open 2BB or limp to try and manage pot sizes--especially if I think the guy is going to pay me off when I start check-raising and re-raising made hands on later streets (I play lower stakes, so there are a few of these types of players).  Against a player that I think folds too much--either to flop c-bets or to certain kinds of bets on later streets, I'm usually opening 3BB with almost every hand.  I will also cut back to 2BB or 2.5BB if I'm playing against a short-stacker (say, less than 50 BB stacks). 

My standard flop bet OOP after 3-betting pre is 2/3 to 3/4 pot.  I do this with almost 100% of my hands when the flop isn't paired or three-cards-to-a-flush (with the rare exceptions being Axx or Kxx flops against players who I know aren't calling 3-bets without AAxx or KKxx).  I play paired flops and three-flush flops a little bit differently.  Here I will sometimes c-bet with 1/3 pot--my thinking is that if I do this frequently with flopped trips/boats as well as with nothing, it's just a good play because aggressive players will re-raise and I can get it all in, sometimes villain will fold, and, in any case, I can get a lot of information without risking a lot of money.

I realize that my turn-then-river play is pretty binary, so I won't try to answer the question.  Pretty much if I'm calling your turn bet of any size OOP, I know exactly what I'm doing for any card that might come on the river.  Which isn't very good--I'm probably giving up a ton of value--but I'm nitty and trying to figure out ways to play differently.

I'm pretty nitty, and my opponents usually pick up on this.  So I actually can get away with checking the nuts OOP for a check-raise quite often--I do this a lot when I pick up in the flow of the game that the villain has decided that a turn bet and a river check mean I've given up on the hand (he is often right).  When I feel that leading the river is right, I vary my bet sizing, but mostly based on how I think the opponent will view my bet.  If I think he will view a half-pot bet as a "feeler" bet to be raised, if he has caught me making a 3/4-to-full PSB on a bluffed draw, etc., all affect what I decide to bet.  But again, I think most players view me as nitty and as someone who very often will shut down the river, so I get the chance to check-raise very strong hands quite a lot.

I default to opening 3x, and then adjust depending on the opponent.  If he's 3-betting a ton, then I might go to 2x or 2.5x.  I don't really want to open less buttons though, but I may also start folding 5-10% of the very worst hands.  If he's very shortstacked that's essentially just an argument for opening small with weaker hands and 3x with strong hands.  I mean, what's he to do about it??  ;-)

In 3-bet pots you can cbet smaller in general, but oop I guess you want to bet bigger in general.  I usually just size my bets according to board texture.

Your turn and river questions are so hand and opponent dependent, it's difficult to really answer them generally. 

What you really want to do though, is look for timing and bet size tells in your opponent.  Look for patterns where he's always betting or raising, or spots where he's always folding.  These questions will be more important in a HU match imo, and they need to be constantly reevaluated.