pretty much anything will bow out and eventually break if it's not given proper support to relieve the pressure. it helps to watch the water pressure and ensure you don't overfill it. most large tanks have a piece in the centre to ensure the pressure is distributed evenly, you can maybe rig something up similar to that.
and yes, suction cups do work
a little heat won't hurt, just make sure it's got good distance from the surface and doesn't get too hot. rubbermaid containers are generally very thick plastic, but it's still susceptible to burning. basically, i would do a bit of testing before i thought to put the little ones in there.