In FTL, the "Save & Quit" options is disabled during combat. If I am in the middle of a battle and have to quit, will I lose all my progress, or can I still continue from some recent point, like right before the battle started?
Your save and quit is a one shot deal. It works by writing your current game data to a file called continue.sav in your FTL directory.
This file only exists if you do "Save & Quit" from the menu. As soon as you load up your game and continue your quest, your data is loaded and the file is immediately deleted. So any interruption in the application, like quitting in a battle, while playing before another "quit & save" will result in a permanently lost quest.
You could always back up continue.sav, but this is considered cheating by many and not the intent of the developers. But it's your game, do as you wish.