The difference in your first pair, with try, is addressed here.
Try to VERB means “attempt to accomplish the action of VERB”.
I tried to open the door = I attempted to cause the door to become open.
If the attempt succeeds, the action of VERB is accomplished: the door is open. If the attempt fails, the action is not accomplished: the door is still closed.
Try VERBing means “perform the action of VERB to find out if it will accomplish some other purpose”.
I wanted to clear the smoke from the room, so I tried opening the door = I opened the door in order to clear the smoke from the room.
The action of VERB is presumed to be accomplished. If the attempt succeeds, the purpose is accomplished: the smoke is cleared. if the attempt fails, the action is accomplished but the purpose is not: although the door is open, the smoke remains.
In both of these, the clause with VERB is the complement of TRY. This is not the case with STOP: STOP only takes gerund complements. However, it may take an infinitival adjunct describing the reason for stopping:
I stopped seeing him = I ceased to see him; I no longer saw him.
I stopped to see him = I stopped in order to see him; I ceased doing what I was doing or I made a stop in my trip so that I might see him.