There is a difference between "sound proofing" and "acoustic treatments". Sound proofing is much harder and more expensive in the end to do.
In a basement, your main problem is (excuse the pun...) "head room", literally. You cannot raise the ceiling, so you have to create an air pocket that is lowered from the ceiling in the basement, which starts making the room indeed very short!
You could go a long ways by affixing a dense material like particle board first, resiliant strips on that, then 5/8" sheetrock. It would probably be good to insulated between the particle board and the upstairs floor with rock wool as thick as you can fit in the cavity between the floor joists. Rock wool has the sound absorption of about twice it's thickness compared to fiberglass insulation, but will cost about what twice it's thickness in fiberglass would cost per square foot.
Without the air pocket, and dense layer, smaller air pocket, then drywall, you will probably contain little sound. Air and density is your friend in sound proofing. Also, you may have to investigate possibly using a double door going downstairs. Ventilation/heating ducts would need to be wrapped in thick rockwool as they will carry sound upstairs quite easily.
As for that garage. The suggestion about building a room within a the garage is the ONLY idea that is going to work. You MUST build a ceiling too. Raising the floor 6" wouldn't be such a bad idea either. If you use staggered stud design in building the walls, you will go a long way towards eliminating transference via the wall studs. Staggered stud design is easy enough. Use 2X6" "caps" for the walls (bottom and top) and 2X4" studs. You would space the studs every 8" on the "caps", with every other stud being flush with either side of the wall. This way, the drywall on either side of the wall will not share any studs, thus significantly reducing transference via the studs. Cool eh? You could use wider "caps" to create yet a bigger cavity between the two walls, which will lower the frequency that will get "trapped" in the cavity if you choose, although, depending on how much space you have between the existing garage wall and the new wall, this may be unneccesary. Make sure to insulate the cavity of the new wall, and certainly it won't hurt to insulate the existing garage walls too, and if they are not finished in drywall on the inside, they should be. Remember, the more dense layers and air pockets you have, the better absorption you will get!
I watched a friend try to "sound proof" a brand new garage he built about 5 years ago where he didn't follow ANY of this advice. The first time his band practiced there, the cops showed up from a complaint from somebody a block away! You could not hear anything about about 400Hz outside the garage, but you could hear 100Hz over a block away quite clearly! Too band for him. His band don't practice there anymore, and he is the owner of one very nice storage space with no bay door.
Good luck.
Creepy