If I tap a land, end my turn, and then play an instant allowing me to untap it again, can I tap that land for more mana
Yes, the only restriction is that the land be untapped when you want to use its ability, it does not matter when you last tapped it for mana. You could even play the instant on your own turn (using the land twice in the turn).
when does that mana get used up
It doesn't matter whose turn it is, each players mana pool empties at the end of each step and phase.
Note that brown "mana pools empty" throughout.
Basically, if I tap mana on my opponents turn, when does it leave the mana pool
You don't tap mana, you tap land. Mana is the result of tapping the land. The way I like to think about it is mana is something I have to spend now while lands are something I can tap whenever you need them (giving you mana of course).