When exactly do you plan to go? If you head over there in mid-late June, you should be completely fine without reservations. High-season for European trekkers doesn't start until much later in the summer (around mid-July), and the trails and huts will be significantly less crowded earlier in the summer.
Otherwise, I would strongly suggest booking your huts in advance. Hut proprietors on a route that receives as much trekker traffic as the Haute Route are definitely accustomed to receiving advance inquiries. I've completed the Haute Route, the Tour du Mont Blanc, the Alta Via 1 (Italy), and the GR-20 (Corsica), and I benefited from reservations on all of those treks, save for the GR-20. Without the benefit of the reservations while trekking along the GR-20, I found myself completely shut out of certain huts along the way and thus forced to sleep on the rock-hard Corsican ground. I would suggest purchasing a guide book like this one, which will provide you with the necessary contact information (including websites) for the huts. From my experience, the best approach is to book about a week or two in advance when you have a better sense of your plans. In terms of weather, that's obviously a tough call...but, as I recall, I didn't encounter much difficulty in sticking to my itineraries on any of those trips. Like any other mountainous region, the Alps are liable to receive afternoon thunderstorms (and possibly a bit of snow here and there), but the storms usually pass pretty quickly.