1: I don't have much experience with Adam's line of polishes, so can't help you with that. I personally use Meguiars M205, and LOVE it. It removed the majority of swirl marks on my Avalon with a Lake Country light green pad. It also took out some pretty nasty severe water spots on a Nissan Murano I worked on. For deeper RIDS, and more neglected paint a two step process would be better; M105, followed by M205. If this is your first time polishing, I wouldn't recommend using M105 as it is an aggressive compound. M205 should do the trick.
2. Polish all painted surfaces. I'm pretty sure the factory spoiler is painted? An all purpose cleaner followed by a trim restorer such as Poorboy's World Trim Restorer is excellent for black plastic trim.
3. Lake country pads are awesome. Don't get anything bigger than 6.5. I have 6.5 and kinda regret it after reading reviews of people who get better results from 5.5. I would say get orange, light green, white, and black or blue. Two of each just in case one falls apart after a while. If this is the only car you will be detailing, then you'll be fine with just one each. You may not need to use orange unless you're have really bad oxidation or defects.
4. Get a nice large waffle weave microfiber towel or drying. Get many many other mircrofibers for different purposes. I don't really have brand preference, just whatever seems like good quality to me.
As always,
Always perform a test on an inconspicuous area first!!