There's no historical precedent for simultaneous launch, but some early space program rendezvous experiments had flights launched somewhat close together.
Gemini 8, 10, 11, and 12 each launched within a couple of hours of an unmanned Agena spacecraft for a rendezvous exercise. They would launch from separate pads at Cape Canaveral, LC-14 for the Agenas and LC-19 for the Geminis.
The Apollo and Soyuz craft used in the ASTP rendezvous mission launched about 8 hours apart, but obviously from very different launch sites.
All the other rendezvous missions I'm aware of launched days apart.
That said, I don't think it's completely infeasible if there were a compelling reason to do it (but there isn't; safely rendezvousing in space is going to take hours whether the ships are launched simultaneously or not). Of your concerns:
Direct interference between the launches: probably not a problem; the big launchpads at Canaveral are something like a mile apart, and the pressure, heat, and vibration from rocket A are way, way, way more troublesome for rocket A than for rocket B.
Exhaust disturbance in the air: probably not a problem for the same reason; if the craft were close enough to one another for exhaust plume impingement to be a concern, the chance of collision would be a far more significant concern.
Command/coordination issues: probably one of the big reasons it's never been done. A lot of activity has to be coordinated for a rocket launch, and doubling that workload without a damned good reason is pointless and hazardous. According to this Q&A though, the infrastructure to support simultaneous launch exists; the redundancy is mainly there for testing, training, rehearsal and maintenance reasons.
If NASA was determined to do a simultaneous launch of two shuttle-sized rockets it could probably replace a couple of the smaller, idle launch pad complexes at Canaveral with one big one, in order to put more distance between the launch pads. That could get maybe 8 miles of separation between the pads, but I don't think NASA would ever be comfortable with the collision risk implied there.