This is called "hidden city ticketing", you'll find quite a few questions on this topic. I'm pretty sure there is a canonical question and answer about this somewhere on the site with all the gory details, but I can't find it.

Book C->A->B and turn up at A and tell the checkin staff that I came there through alternate means. I will obviously have to be before the flight from C departs to avoid being a no-show.

Almost certainly no. Most fare rules and/or conditions of carriage specify that segments must all be flown in order. Indeed if you didn't show up in C in time they would just cancel all segments. Not sure they would go around that in the scenario you give, and depending on the airline and airport, you may not even be able to check-in that early in A anyway (depends on the exact timing).

Of course, airlines and staff may have some leeway, but it's nearly impossible to guarantee in advance, so you risk losing the whole round-trip at the last minute. There are probably exceptions for the more expensive "flexible" / "full fare" tickets, but you would have to double-check that, and since you're trying to save money I don't think this is an option for you.

On the way back from B -> A -> C, can I disembark at A after requesting the checkin staff at B to book the baggage only until A?

In theory no, though I've seen exceptions, though with a quite more complex scenario where the passenger definitely wasn't trying to trick the airline (and it was obvious). Again, nearly impossible to guarantee in advance, but it doesn't hurt to ask at check-in time.

This is called "hidden city ticketing", you'll find quite a few questions on this topic. I'm pretty sure there is a canonical question and answer about this somewhere on the site with all the gory details, but I can't find it.

Book C->A->B and turn up at A and tell the checkin staff that I came there through alternate means. I will obviously have to be before the flight from C departs to avoid being a no-show.

Almost certainly no. Most fare rules and/or conditions of carriage specify that segments must all be flown in order. Indeed if you didn't show up in C in time they would just cancel all segments. Not sure they would go around that in the scenario you give, and depending on the airline and airport, you may not even be able to check-in that early in A anyway (depends on the exact timing).

Of course, airlines and staff may have some leeway, but it's nearly impossible to guarantee in advance, so you risk losing the whole round-trip at the last minute. There are probably exceptions for the more expensive "flexible" / "full fare" tickets, but you would have to double-check that, and since you're trying to save money I don't think this is an option for you.

On the way back from B -> A -> C, can I disembark at A after requesting the checkin staff at B to book the baggage only until A?

In theory no, though I've seen exceptions, though with a quite more complex scenario where the passenger definitely wasn't trying to trick the airline (and it was obvious). Again, nearly impossible to guarantee in advance, but it doesn't hurt to ask at check-in time.

In your specific case, since B is outside the US and A and C are in the US, you have to go through immigration and customs at A, so you'll get your bags there whatever happens, provided there is no US pre-clearance at your point of departure. You are then supposed to drop your bag back into the system right after customs, but if you ignore that and exit the airport there isn't much they can do (other than flag you and make your life miserable in the future, but I suppose they only do that for repeat offenders).