So, player 1 bets the big blind, followed by players 2, 3, D, and then S puts in the other half of the blind (he needs to put in the other half to call).

Aha, I get it now. Everything for that betting round counts, even the blinds count as part of the bet. So all that's happening with a raise during round of betting is that the minimum is raised and everyone has to up their bets to that minimum.

So after the flop, then again after the turn, is the minimum that new amount or does it get reset to the original minimum?

Okay then how do checks work? Check is to pass without betting yet remain in the game. Does that mean that everything after the flop is a raise? Let's say after the flop, first-action-guy says "check" and everyone does the same. Okay, we see the turn with no new chips on the table. But what if first-action-guy says check and the next guy in line puts 10 into the pot. Is that technically a raise which everyone else has to now match?