I was actually surprised by the complete lack of insects throughuot the entire walk. I only got bitten twice during a short day hike around Chamonix, but never anywhere else. You don't need any bug netting.
There were a few days in August which were really quite cold, the ground and my water froze some nights. Snow fell quite often at around 2,000 meters, so if you were planning to camp along the higher, alternative trails in October I would definitely say that you need to bring warm gear. October is probably still walkable, but you can probably expect snow quite often.
I wouldn't say that camping is really difficult along the TMB, though the Italian section I found to be the most strict about enforcing the rules. Lots of helicopters flying around constantly, watching the trails. I do agree with Roger about people preferring that you stay at the gites and refuges. I camped about 3/4 of the time, and everywhere I camped I always found others in camp with me. So the camping is possible. The only big problem you will have in October is being able to find food and water supplies, because a lot of the mountain huts will be closed.
Technically, unless you take the alternative routes, the TMB is a really easy trail to walk (much easier than walking here in the Japan Alps) with only a very few sections that are actually potentially dangerous (in terms of rock fall or uncertain footing). Trail signs are for the most part extremely clear and most of the trail is so wide and well-trodden that ultra-marathoners run the trail at night. A few sections follow paved roads.
If I would do the trail again in August I would definitely go with a tarp and bivy, instead of the tunnel tent I used (though I occasionally appreciated the extra warmth it provided). I wouldn't go with a tarp and bivy in October. There were some sections (notable the stretch between the Col du Bonhomme and Rifugio Elena) that were quite exposed and very windy so they would tax a tarp unless you have experience and confidence with using a tarp in very adverse conditions. Also, though I don't know about weather conditions for the Alps in October, the very common afternoon violent and cold thunderstorms must surely have a counterpart in autumn. I would say be careful with your shelter.
As Roger suggested if you really want to go light you can completely forego camping and stay in the small towns or mountain huts along the way. You will reach some settlement at the end of each day; this is not real wilderness. Most Europeans go this way, so you rarely see anyone with a big pack.