The heat gain is lower if you don't kill. but you do still gain heat. I'm fairly sure it is less again if you stealth exit rather that looting ammo etc. If you are looting, successfully intimidating bystanders reduced heat and rep damage, but I think failure makes it even worse. If you are working for Brave Star, make sure you speak to their ladder connector Texas Powers (or both if you are playing the tutorial - Lt Simon I think, you get introduced to him as part of the story) There are some pretty good rewards.
I've never done much on the matrix so can't help you there.
One other thing, sign up for an account on this board. You can then get alerted when any of use reply to your questions.
You can also drop heat by sitting in a restaurant and buying refreshments. Each time you do, your heat will drop 0, 1, or 2 points. But it also uses a lot of time up. Sleeping in a hotel is more efficient if your heat is low enough to dissipate with a single night's sleep. But that only works once; you have to find another hotel to drop heat again. If you have so much heat you are afraid to go looking for another hotel, but have cash, go to a restaurant. You can spend as much time there as you want, paying cash to drop heat.
Sometimes you gain heat even when you don't have encounters. The amount of heat you drop when completing a mission is somewhat random. At the completion of the mission, sometimes you drop more heat than you gained by taking the mission, sometimes you drop less.