No the run does not count. An out at any base would be a force out and no runs can score,
If the runner scored and the third out is made because a runner is called out on an appeal play such as failure to retouch on a fly ball out or a runner missing a base the run would count
If bases are loaded and it is ball four, yes. Otherwise, no Rule 10.04(a) explains part of this answer.
Yes, a batter gets a RBI through a Hit-by-pitch, walk, or hit if a runner scores on the play due to their plate appearance
Yes. There are only a few times that a batter helps a teammate score but does not get an RBI. Example 1: Bases loaded, 0 outs, batter grounds into double play 2nd baseman to first baseman, runner at third scores. The batter is not credited with an RBI. Example 2: Runner on 3rd, 2 outs, batter hits a routine groundball to shortstop, but the shortstop makes an erro. The batter is not credited with an RBI because if the ball was played cleanly the run would not have scored.
Batter is safe at first base, runner on third scores a run.
No, it is scored as a ground out and the batter is credited with an RBI.
Yes. If a runner crosses home plate before the third out is made (unless it's a force out), the run counts. For example, say the Yankees have runner on third with two outs. The batter hits a ball to the gap in right center. The runner scores, but the batter is thrown out at third trying to stretch a double into a triple. Since the runner on third crossed home plate before the batter was out at third, the run counts. On a force play (at any base) the runner would not be allowed to score even if he crosses the plate before the force is made.
It is only a sacrifice bunt if someone scores. If no one scores, then the player who bunted isn't credited with anything.