I got a lot of money from the first heist mission in GTA V. Is it possible to play it again and get the same amount of money and/or level up the abilities of my crew?
Navigate the start menu, and you can replay a mission. When you choose to do so, the game will notify you that it creates a save state of your current progress, allows you replay the past mission, and then restores that save state once the mission is completed.
You can't gain additional money, or level up your crew as nothing is saved from replaying missions. You can, however, do the heists the second way (as most of them give you two approaches) to attempt the gold medal objectives, which will save if you achieve them.