FWIW, I've been asked many times to tidy up after this kind of problem, and not once (in a case where the customer made the initial contact like this) have I ever seen any significant malware on the machine. The odd toolbar, or junk "security" software yes, but never real malware. However I have seen it from those bastard scammers who cold call you.....
The advice given is correct - get it off the net, get it checked out and preferably reloaded. However I would suggest the probability of a problem is lower than you suspect. However the possibility is still there.
As for the remote control software, I've yet to see any of these remote repair guys install anything to worry about - in every case the software has been passive, not active, requiring action by the user before the call centre makes a connection. I really doubt if its set up to be malicious. However.......I could be wrong