war hospitals do not help with natural disasters, only field hospitals do (maybe they help with "unnatural disasters" but not with regular ones)
Best way to prepare is to maintain about 10% of the maximum amount that you can stock in the "set up resources for natural disasters" or something. It's under the disasters menu.
I've noticed one thing, namely that after the disaster has been "handled" my population had increased to more than it was before, plus the 20k per month average growth rate.
I was also informed by some of the veterans, that disasters are GOOD! What you do is take over a C3 when u get hit with it, then have that C3 set up every single team it can, then ship them over to the hit country. After all those resources are shipped over, unpack those teams, then set up some more in the C3, and repeat. Simple way to strip the population from a C3.