Are we obligated to advise employees that this is being done?

Are there risks to  firing employee after he monitoring on the basis of the content of emails?

You should have it somewhere in your company policy. However, it is a common knowledge or reasonable man issue. Everyone should know that if you use internet and/or email at work you have no any privacy rights. The employer has the right to monitor your activities on their computer. And the right/duty to terminate employees if you are mis-using their equipment.

Make sure you know your state law in this area.  Some states require advance notice to employees before you monitor their activities.  In any event, having a policy stating that computer use may be monitored is a good idea.