If your ATM card has a Cirrus or Maestro logo on it, you will definitely be able to find a cashpoint that takes it. Burma is the only place I have ever been in many years of travelling that doesn't have cashpoints.
Don't think most travel insurance covers cash, sorry.
If you were in the deepest of shit, you could always get a cash advance on your credit card over the counter at any bank.
Depends where you're going, really. In Shanghai you could walk around with a diamond tiara and US$100 notes stuffed in your pockets and nobody would bother you, but somewhere like Phnom Penh is a completely different kettle of fish.
If you are going to use your card, tell your bank before you go. My sister was on holiday this summer and tried to use her card, the bank stopped it because of unusual activity i.e she usually only uses it in the UK. It took a day to get it sorted out.