How tight should a screw be screwed in?
Screwing in to drywall is really the only thing on your list of questions to worry about. A drywall screw should be screwed in to where the head is just barely under the paper on it's face. They make drills, and drill accessory bits, specifically for this task.

For everything else, run the screw in until your material is tight.

As for what tools to use... This is actually personal preference. I personally use an impact driver 95% of the time. But, I've been building houses for 30 years. An impact driver, compared to a drill driver, is better for driving screws in to dense materials. A drill driver will work also but requires more force on the tool to get the results of an impact. For someone inexperienced, I'd recommend getting use to using a drill driver, utilizing it's different clutch settings, before moving on to an impact driver.