Even though you have chosen not to, in this instance, it's probably a 'good idea' for anyone either using computers in an office environment, or lent equipment for work use, to check out company policy on IT usage.
While I'd always suggest someone get a free e-mail account for personal mail (rather than give anyone in a social situation their work e-mail), mainly for the individual's privacy (there are commonly policies saying that all work mail can be examined - whether for compliance {eg giving a client advice, and the e-mail may become part of a dispute later} or simply to cover for someone being unavailable because of holiday, sick leave, etc, and someone else needing to see what comes in.
When it comes to using the internet, all too many businesses are vague about what is allowed. Clearly, there could / would be a ban on anything which disrupts the work environment (some online 'game' for example, if it has sound effects, or playing music/ radio though with headphones, I'd expect it to be OK), or viewing obscene material.
However, what one is allowed to do during a coffee break, or lunch hour, needs to be defined. I'm remember someone went to an industrial tribunal after being sacked for booking her holiday during her lunch hour. Unfortunately, despite no clear policy, she lost the case.
(Just did a check and it was in June 1999. Still 'fresh' in my memory. Oh dear! I didn't remember her name, however - a quick search for 'Lois Franxhi' will find the story on the BBC website.)