The answer to your question can be found in a reading of Rule 11 and the definition of the Teeing Ground in the rule booklet. In summary the teeing ground is a two club-length area that is measured by the front and outside of the two tee markers. A ball is outside the teeing ground when all of it is outside the teeing ground. In other words the ball must be teed within the two club-length area to begin the play of the hole. The player need not be standing within that two club-length area when starting the play of the hole. Should the ball be outside the designated area Rule 11-4 clarifies the correct procedures in the various forms of play.

This is not uncommon when a cart path is adjacent or close to an out of bounds fence or wall. The first thing to remember is that interference by an immovable obstruction occurs when a ball lies in or on the obstruction, or when the obstruction interferes with your stance or the area of your intended swing. If you find your nearest point of relief as described above, then you must drop the ball within one club-length of and not nearer the hole than that point. If the ball when dropped comes to rest nearer the hole or rolls back into the condition where interference still exists, you are required to re-drop. (See Rule 20-2c regarding re-dropping procedure).

Scenario 1
You are not allowed to play a provisional ball for a ball that might be unplayable, therefore when you dropped and played another ball, that ball became the ball in play under the stroke and distance option (Rule 28a). When you holed your next stroke your score for the hole was four.
Scenario 2
If you thought that your ball might have been lost outside a water hazard or out of bounds, you were entitled to play a provisional ball. However, you had to tell your fellow-competitor or opponent before proceeding. If you failed to do so and played another ball, that ball became the ball in play under penalty of stroke and distance. The original ball was lost. (See Rule 27-2a). Your score for the hole would have been four.
Scenario 3
If you had proceeded correctly under the provisional ball rule, (Rule 27-2), and the original ball was found before the provisional ball was picked out of the hole, then you were obligated to complete the hole with the original ball.
Several Decisions relating to this question: Dec. 27-2a/2 and Dec. 27-2b/2

You have stated that your fellow-competitors and you all agreed that your ball was in fact in the bunker and not retrievable. The Rule therefore, provides that since you opted not to play the ball as it lies you had 3 additional options:
1) drop a ball in the bunker without penalty at the nearest point, not nearer the hole, where the depth of the casual water is least-(Rule 25-1b (ii) (a); or
2) drop a ball behind the bunker under penalty of one stroke-Rule 25-1b(ii)(b); or
3) deem the ball unplayable and proceed in accordance with Rule 28.
Hope this helps to clarify your question.

The answer depends on whether the player authorized the caddie to repair the ball mark. If the player authorized this action there would be no penalty. However, if there were no player authorization then the fellow-competitor would incur a two stroke penalty under Rule 1-2. Handling this situation should be done in firm but diplomatic way.

Yes. The ball's location does not affect a player's right to remove sand on the putting green.

Under Rule 28c, a player may drop a ball within two club-lengths of the spot where the ball lay, but not nearer the hole. If a player drops a ball and it rolls and comes to rest less than two club-lengths from where it first struck a part of the course, the ball is in play.

Under Rule 20-2c, when a ball comes to rest in a position from which relief was taken under Rule 24-2b (immovable obstruction), Rule 25-1 (abnormal ground conditions) or Rule 25-3 (wrong putting green), or rolls back into the pitch-mark from which it was lifted under Rule 25-2 (embedded ball), a player must re-drop.