1. No, it’s incorrect. It should be:
我在冬天去爬了山。
Verb+了+noun 爬了山,吃了饭,刷了牙 etc
2. 在毕业以前 before graduation; 在……以前/之前 before…
3. 赢 means win. 球means ball. 赢球 means win or win the game.
4. 早上 is colloquial expression. 早晨 is the written form. Usually we say 早上好 (good morning) instead of 早晨好。
5. They are interchangeable.