Sorry beforehand if this is a really obvious question! If a ball is caught in foul territory (batter is out), is it still live and anyone on base can tag up and take off?
Yes, the caught foul ball is live and players can tag up and advance at their own risk.
That is why there are some situations (Tied game, runner on 3rd, bottom of 7, less than 2 outs) when it might be better for a player to let a foul fly ball drop rather than catch it.
Yep, live ball. Once the foul fly ball is caught the runner can legally tag up and advance a base. I can remember a couple of times where we've screamed at our outfielders to let it hit the ground.....do not catch it!! And both times it worked in our favor.