It's fine to fill screw holes and other small wall dings with patching compound, but for dime-size and larger drywall repairs, and for holes that are deep, it's best to use a joint compound that sets up by a chemical reaction. These are available in powder form with setting times ranging from five to 90 minutes.
Can you put drywall on top of drywall? You can but in order to make it look right you should pull the window and door trim and add jamb extensions before you reinstall it. Similarly you should consider adding electrical box extensions.
Generally speaking, if the damage to your drywall is minor, such as a small hole or crack, you can probably repair it yourself. However, it is best to hire a professional for more significant damage.
You can save a lot of money by installing drywall yourself, but it's not usually a job for a DIY newbie. The cost of purchasing the panels (without the additional labor and finishing costs) is around $0.40 to $0.65 per square foot, so about two-thirds less expensive than professional installation.