What can you do? Can you sue the debt collector? Can you sue your
employer for improperly cooperating?
Yes. Also report the debt collector to the appropriate regulatory agency for violation of laws relating to debt collection. Possibly refer the debt collector and/or employer to law enforcement and the DA for theft and fraud.
Honestly, this is a pretty rare fact pattern. Far more common is for the debt collector to have a court judgment obtained by default against the debtor and for the debtor to not realize that the debtor has been served with process or for the debt collector to have used "sewer service" in which the process server lied about delivering process to the debtor.
More to the point, what needs to be done to halt this process until
the matter is sorted out?
Probably it could be stopped with a temporary restraining order from the court in which the lawsuits are filed. But a letter to the employer and debt collector pointing out the there is no court order of judgment would probably stop the employer from coooperating.
Is is possible to accomplish this pro se, or would you need a lawyer?
You need a lawyer. This is very hard to do right pro se.