What happens to the FTL progress when I quit during combat

By "combat", they mean man-to-man combat, not ship-to-ship combat. Basically, Mantes make good raiding parties or security guards.

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).