1. Yes. I use a mix-stir, and it works quite well for me. Shaking will work just fine, but it is too hard on my back.
2. Absolutely. Pitching healthy yeast will do wonders.
3. A swamp cooler has worked very well for me the last few years, but I also like my house temperature to be at 60F. I decided this year to ramp up my brewing during the winter, so as to make good use of the naturally lower temperatures, then take the summer off. My beer made during the summer just hasn't been that good, and it has been too much of a hassle to try and keep the temps steady.
4. Yes, I think getting a water report, and using bru'n water has resulted in the biggest improvement of my beer.
I think you have a great idea of were you want to go, and I would try as many as possible, as soon as possible. But there is something to be said for one step at a time. Getting the ph right for the mash, and pitching healthy yeast has resulted in a big improvement in my beer. Before I got those things right, I also brewed enough to learn my process, and figure out my equipment, which I feel to have been an important towards working out the kinks.