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.
They die. Same when the ship is destroyed when they are still boarding it. I have tried it sadly. :(
The game does warn you if you try to jump away before you gather your crew. But the game doesn't warn if you teleport crew unto a ship without life support.
Your crew is also gone when the enemy ship jumps away.
Best Answer
Yes, you will be able to gain new crew members. When you get a crew from an event, e.g., an interface will appear at the end of the event showing you every new crew member + the new guy (who will helpfully be flagged as "new"). You then have to chose to dismiss a member since you are over the limit. If you see crew for sale in a store, you can go into your ship screen and dismiss a member there so that you can buy a new one.
In the comments, ChargingPun confirms that when you are on the "crew dismissal" screen, you can still mouse over crew members to see their skills.
Also please note that as of patch 1.5.4/the recent add on, potential crew you encounter can come with skills which have gained experience. It is possible to get a crew with max [insert skill]. This is tremendously useful, as it both allows one to more easily make up for later-sector mistakes as well as maybe find someone with decent shields skill (heh).