Sounds very questionable to me especially since they said they didn't know if they have a website, and then when you were asking legitimate questions they claimed you were being difficult so they would serve you at work.
I would say they are either tyring to:
a) Scam you into giving them money or, possibly even more damaging, get some of your protected private information to commit ID Fraud.
b) Disreputable collector trying to scare you into paying for a debt that is way past the Statute of Limitations if it was really ever your debt all, which seems like it was not, or someone else with similar name has a debt and they picked your name first out of a list of possibles.
Seems to me they realized you weren't falling for their tricks so they used the "serve you at work" line to get you off the phone quickly.
If they call again I would not give them any personal informat such as SSN, Acct #s, Phone #s etc..Then tell them that you are formally requesting that all further communication from them be in writing.
