Please send us the hosting specs so I can see if there is anything suspicious there.
Regarding the session and MonoX, you can safely use the Session in MonoX, what I was thinking is that you may be using WebFarm or some kind of cloud that can trigger that kind of exception if you didn't handle "cloud" scenarios ?
Could heavy reliance on the Session be an issue with our web parts and MonoX? I'm thinking no, but though it wise to ask now.
Regards