Legal Question in Criminal Law in California

Yes, assuming someone is suing the accused criminal for the conduct that underlies the criminal case the two matters could proceed concurrently, and it would also be to the advantage of both parties to the civil case to request that the civil case be stayed until the conclusion of the criminal case. If the defendant is convicted of the felony, the civil case against him becomes a slam dunk, and it is also entitled to trial setting preference (the plaintiff's attorney would have to file a motion for this.)

Of course. There may be logistical problems, but nothing stops either from proceeding.