Hi there " iglehart332", and welcome to the forum!
Why cant I just use screws to attach the drywall
That would be almost the same as gluing the drywall to the underlying USB: it would prevent them from acting independently as well as together. The net result would be isolation that isn't as good as it could be, especially in the low frequencies.
Also I hear a lot of people talking about using cleats to hold the drywall in place. What is the purpose of this?
Same as furring strips: they hold the drywall in place around the edges, keeping it pressed up against the OSB. Different name, same purpose.
3. What size gap should I leave between the studs and drywall? I've read a lot of debate on this. Rods book says 3/8", some say 1/4", some say none. I'm confused
Well "none" is definitely not the right way! You do need a small gap to ensure that the caulk seals around the edges well. 3/8" seems a bit large to me, but if that's what Rod recommends when you do it his way, then that's the way it should be done. 1/4" is about as small as you'd want to go: any smaller than that, and you can't be sure that the caulk is filling the gap well.
4. Are their any benefits of using acoustic caulking at this phase or can I just get regular 25yr white silicone caulk?
Acoustic caulk is specially designed to do the job as well as possible. Bathroom/kitchen caulk is a good substitute, yes, but its better to use the right stuff if you can. If you do prefer bathroom caulk, then make sure it is the type that never hardens, and always remains flexible and rubbery.
it would be easier to remove the vinyl siding and add some sheets of OSB to the outside and then replace the siding...
Yup, it sure would be easier! Quicker, less expensive...

If you do that, use 5/8" OSB for the new layer, and stagger the joints such that the joints between panels on the new layer do not line up with the joints on the existing layer. And of course, seal all gaps as well as you possibly can.
- Stuart -