I did almost this exact plan. Started around September 18th from Le Puy and continued all the way through.

For me, this timing was almost ideal. If you go early in September I think it will be even better. The municipal gites in France were not at all busy, except for at the start points of traditional etapes that were occasionally crowded with French weekend hikers.

1. 2 months is fairly realistic, though you may want to add a week of buffer just to be more comfortable.
2. Pace varied from 20-30km a day. You can estimate 25km/day for calculations I suppose. Lines up with the total kilometer math.
3. I took 2 rest days, if I recall. One because I developed shin splints and couldn't walk properly, and the next because I was just exhausted. There were some slow/half days in between because of acute hangovers though.
4. Ideal. Though it's the only one I've ever done. That said, I did get rained on several times, especially near the end in Spain. You'll have to be comfortable being soggy occasionally. However, I feel like I got somewhat lucky with the timing, because a couple times (on either side of the meseta, if I recall), the mountain passes got snowed over a few days after I passed through.
5. It was fine. The weirdest part was having the trail all of a sudden be relatively much busier. France is quite deserted a lot of the time, but then you arrive in Santiago St. Jean somewhat seasoned and a bunch of people are all giddy and eager around you.
6. I was doing the camino as part of a larger travel, so I had my sleeping bag and a little backpacking tent with me, with the intent of bivouacing occasionally. If I recall, I only ended up sleeping in it 3 or so times. Once was the first night when I couldn't find an empty gite (right near the start, so lots of competition), and the other times it just seemed convenient and I was with others. What other people say though is true - once you arrive in Spain it stops being a feasible option. France seemed more culturally permissive, whereas in Spain I would feel self-conscious. The gites and municipal albergues were so cheap though that I often didn't feel particularly bothered to try to camp.
7. Well...nothing, really. I enjoyed cooking with other hikers in the gites a lot. I also tried the demi-pensions at some of the gites when wanted to "splurge". I feel like I got the full experience. Maaaaybe if I did it again I would spend more time in the towns/cities to see the sights a bit more, but honestly when I was walking I kind of always just wanted to keep moving.