When boarding, I tend to aim for the Oxygen subsystem room - it's typically small, and critical enough that I will quickly attract attention. If they don't respond, then I've pretty much sealed their death quickly.
Meanwhile, I'll target whatever critical system I choose (usually weapons, shields, drones, or engines) with my laser weaponry. Once I've hurt one or more of their crewmembers, I'll target the medbay with missiles or bombs. That way, I can finish off heavily wounded crewmembers while disabling their ability to heal. Fire bombs and subsystem bombs are the best for this sort of attack.
I do believe that firing on the rooms where your boarders are will hurt them as well as the enemy, so I tend to avoid this. Trying to get the timing right for lasers (to reduce shields) + beams (to do damage) is also kind of tricky if you're also monitoring a boarding party.
Also, it helps if you've got a couple of teams of boarders - while you're healing the first squad, send the second squad in to keep the pressure up.
Some ships are particularly bad for trying to board, however. If they've got significant crew, especially a lot of mantises, I may give up trying to win by wiping them out. On the other hand, getting into an engagement and realizing they have no medbay pretty much means I've got a win locked down.
As you've noticed, Rock and Mantis crewmembers are the best to send as boarders. Don't forget that you can pause the game with the space bar, so if you think getting your crew back on board is going to be a close call, pausing and unpausing to slow the game down may help you slice the timing razor-thin.
Possible spoilers if you've never been to sector 8:
It's also been noted in a couple of strategies for fighting the final boss that the weapon systems are in isolated rooms with just a single human guarding each. If you send 2 of almost any race of boarders into these rooms, you can kill the guard and disable the system, thereby making this fight easier. The best weapon to start with is (in my opinion) the missile launcher, which is third from the left.
There is a developer who has answered that med-bay upgrades do not affect the Engi med-bot dispersal healing speed: source.
I also measured the effect with a fully upgraded and fully powered med-bay. Before any upgrades, it took 10 seconds for a crew member to hit 50% health in front of an open airlock. After all upgrades and med-bay fully powered, it took 10 seconds for a crew member to hit 50% health in front of an open airlock.
Engi Med-Bot Dispersal is not affected by MedBay upgrades.
Best Answer
No, the Engi Medi-Bot Dispersal ONLY heals crew members on YOUR ship. This means they will not be passively healed when boarding enemy ships. Only the crew that is currently onboard your ship receive the passive health-regen.
As RavenDreamer points out, Mind-Controlled allies will not receive the passive effects from the Medi-Bot Dispersal because during Mind Control, the controlled target acts like an "enemy".