I entered a sector where a Federation ship was in trouble but disappeared into some clouds. I chose to follow/help the ship. What happened is that a rebel ship was chasing it, so I had to fight the rebel ship.

I targeted its drone with my missles and lasers and the ship surrendered. The drone stopped, but the rebel ship was just sitting there. I had taken a little damage so I had my guys running around "repairing" some of the damage. Then I noticed... some of my other systems were taking NEW damage! But from where?

First the O2 turned red. Then the Engines. I sent my guys to the O2 room and noticed they actually walked through a room with fire in it. But there was no fire icons visible UNTIL they walked through the room. I managed to recover some of the O2 but my guys were almost dead, so I sent them to Medbay to recover. Then I noticed the Engine room turned red! But again, there was no warnings or indication of just exactly what is happening.

How are we supposed to fight/fix something we can't see? Or figure out what happens? I understand the game is supposed to be "hard", but when the game comes with no manual, and a super-skimpy tutorial, it is just NOT FUN to have things happening with no clues what they are, so you're just supposed to die?

What causes systems to just spontaneously go bad? Was that rebel ship that supposedly "surrended" responsible for the new damage? The rebel ship never disappeared... he just surrendered and gave me some missles, drone parts, and scrap... but still was sitting there. Is he shooting at me? ARGGH.

Sounds like you were fighting inside a nebula. You can tell when you are because the background will be composed of purple clouds. And usually when you're inside a nebula your sensors won't work and you won't be able to see inside any of your rooms unless a crewmember is there. That's how come you wouldn't have been able to see the fires initially, and they would have ended up spreading. Also inside nebulas you can encounter ion storms which might have been doing damage too.

Yeah, fires in a nebula are a pain. Here's a tip, though: regardless of what your sensors are doing, you can always hear when your ship is on fire, because there will be a crackling 'fire' noise played over the background music. Remember to listen for this, especially at the end of a fight, and if you're not sure where the fire is, just vent air from all non-essential rooms until it stops.

No, no manual, just the tutorial (which doesn't cover fire IIRC).
However... if you look at the power bars for each of the systems in the bottom left of the screen when something is on fire/being attacked by boarders an icon will appear just above the affected system and the energy bar will steadily turn red so you can see how far along the damage is.

A Manual? ... hell no. But i recommend watching some lets plays on youtube. like this one http://www.youtube.com/watch?v=3c_QY0dnI-M&feature=related