Automatic logoff; strange cache behaviour

We have implemented our online shop based on nopCommerce. The site is hosted on a shared server, on a service provider; the server is IIS7.
We have a big problem: after a certain period, the logged on users are automatically logged off by the system.
I have tried to change different timeouts from web.config, but the phenomenon appear again.

Another problem is related to cache; we developed some additional functionalities where we are using cache (the class NopCache); after a few minutes of normal functioning, the cache get a strange behaviour: we put an object in cache and when we try to get it, the cache is empty.

I put the site also on one of our servers (also with IIS 7), and these strange phenomena don't appear.

I came back with some additional details regarding the first problem.
I put some logging in Application_AuthenticateRequest (from MembershipHttpModule) and it appear that after some time (seconds, not minutes) HttpContext.Current.User is null. I still don't know why this is happening.