You're going to get various opinions on these questions, so you'll need to sort out most things for yourself and decide what works for you. I've gone to a lot of effort to educate myself on the topic and have done a lot of experimenting, and I'll share with you what is working for me.
I agree with Nammie to 7, the "Thread Therapy with Dr. Bob Educational Videos" from Superior Threads are a good place to start, and they are available to watch from their site on the web, or you can purchase a DVD (right now it is FREE). I'd highly recommend taking the time to watch this all the way through.
1- Should I be staying with 100% cotton for quilting? ~ That depends on whether you want to or not, there are opposing camps on this one. I like either a fine cotton, Presencia from Harriet Hargrave is my favorite, or Bottom Line poly thread from Superior. Both are fine 60 weight threads with very little lint. I started out using any thread that was the color I wanted, but learned along the way that cheaper, heavier threads do not perform as well and can cause all kinds of problems including inaccurate piecing, excessive lint build up and thread breakage.
2- What weight is best for quilting? ~ That depends on what finished result you want. For a fine "heirloom" look that blends in, at lightweight thread is best and again the Presencia cotton works very well. King Tut from Superior is another cotton thread that performs very well. When I want my stitching to stand out (which I often do), I use a poly embroidery thread with a bit of a sheen to it. My favorite is "Magnifico" by Superior Threads. It comes in wonderful colors and I never have problems with breakage. Whatever thread you use, be sure to use a needle that is the correct size for it. I'm 100% sold on the Titanium Top Stitch needles from Superior.
3- Does the thread weight depend on batting or density of quilting? ~ Again it depends a bit on the look you are trying to achieve, but lighter thread will give you a finer look which will work out better for very dense quilting. That also allows you to use a smaller needle so that you don't leave behind visible needle holes in fine fabrics. On the other hand, if you are quilting on a quilt that is thick and heavy weight because of the combination of batting and outer fabrics, a heavier weight thread and larger needle will work better.
These are just my opinions on what is working well for me at this time. I will tell you that when I first started FMQ I thought that there were some threads that just did not work in my machine (It is a Juki TL989Q), but as my skills improved, those same threads worked just fine...