There's not really a timeout such as this that I'm aware of, or at least not some settings to update it. You can check your settings under Manage -> Appliances and check the Login or Settings tabs under Properties. If you're refreshing the entire page instead of the widget then you will likely have to sign in again. To avoid that you can set the account for your NOC view to remember credentials, automatically sign in, etc.
For your second question; LEM is event based. LEM won't be able to monitor accounts for you unless you just want to be notified when the account finally does expire. It sounds like you want to set up a custom widget in SAM to track that from what your description mentions.