But it's also true the other way round !!!! It takes longer to cool a large volume of warm air. So I don't quite agree with that part of what you're saying.
No, you are confused. The purpose of the fans is not to cool the air. They are there primarily to help the air circulate through the case. Their secondary purpose is to exhaust the old/bring new air into the case. The reason that is the secondary function is that this would happen anyway, unless your case was airtight. If you reverse priorities, as you suggest, either every component needs to be in the airflow path (not possible in an ATX setup) or you need so much pressure that it will be very noisy.
As long as you don't have "pockets" of air in your case, and as long as your fans allow for some cool air to get in and for some hot air to be exhausted, there is no reason that a damping material would heat up your case if you don't change your airflow pattern.
Exactly right. However, this means that the only thing you will be able to achieve with your damping material is mass loading. There is no way to do anything about most airborn noise without using a very thick layer of material or blocking some holes -- either of which will effect your airflow.
Again, the argument is not that damping materials do nothing, just that they do a lot less than many people are expecting. I just want people to makes sure they have done things like work on silencing their fans and HDD, before getting too excited about damping.