I have a BES Blackberry with AT&T.
If I purchase an unlocked Blackberry for personal use and use it with the BES AT&T SIM card occasionally, would it work?

I think you would have to "change devices" each time as only one can be active on your sim card at any time. I could be wrong - but that has been my experience with T-Mobile.

The phone will work but you will not get BES email on the unlocked BlackBerry. The BlackBerry PIN is associated to BES.

Can I get BIS and use the Blackberry Browser?

I believe you would have to sign up for BIS for the unlocked phone. I could be wrong so hopefully someone will correct me if I am.

x14 = Bingo

Just a note of caution here. If you have a company supplied BB, I would not try to circumvent their rules and policies and try to put the SIM in your personal device. That is, if you value your job. The best solution is to get your own device and pay for it.