I did the Raid first week of June last year and I've done a fair number of pyrenian and alpine trips previously. I reckon:
1. Probably, and you may not even need a gilet and arm arm warmers! However my current desktop background is a picture of my bike in heavy snow taken at the top of the Tourmalet in July. You have a van carrying your stuff - so take a good top with you, warm gloves, fleecy hat and decide on the day each day what you'll need to carry. It'll be pretty obvious though, the weather reports and conditions are forseeable, you're very unlikely to be caught out if you've talked to people and checked the weather each morning.
2. I wouldnt bother with deep sections, but tubs vs clinchers is your choice. Roads are fantastic (at least compared to the UK); I've never had a puncture in France, Belgium, Spain or Italy! I'd suggest ali braking surfaces, as you may be unlucky and get a wet day or days, maybe even descending in fog or mist which means constant braking all the way down for upwards of 30minutes. Personally I just used my training wheels - DT 1.1 rims 28/32 round spokes on 240 hubs. My thinking is that its not a race, I dont need to gain seconds advantage over anyone (or minutes for that matter); I'd rather have the security that if I broke a spoke I can ride on, and fix it easily myself in the evening, and comfort is more important than performance.
3. Just do what you can do. The first and last day have minimal climbing, so its just three days of cols (how hard can it be?
). Remember that you have all day; IMO its best to use it - take in the scenery, stop on the cols, take some pictures, etc, etc, so its not as tough as it sounds, and dont get stressed about it
4. Recovery - stretch and eat loads!
More generally, for the average rider assuming conditions arent ridiculous (excessive heat or cold aside) its not excessively hard. You get up early, do a 2 hours, stop for your hot chocolate and a snack and take in the view, do another couple of hours, have an hour for lunch, knock out the remainder maybe with stop for a coffee on the top of a col or whatever. You roll into the hotel at 4pm, kip for a couple of hours, have a massive dinner and a couple of beers, go to sleep at 9:30 for 10 hours, wake up refreshed - and rinse and repeat! Its not like setting a time on the Marmotte where you end the day shattered, or at least I wouldnt approach it like that.
I had phenomenally crap weather - pouring rain from start to finish and thick mist on most of the cols. I knocked out day 4 in one hit, treating it like a race or event. Its doable, but there's less enjoyment - I cant recall anything from that day apart from keeping my head down, whereas I can remember having a chocolate (in the rain) on the aubisque, dropping off down into argeles for a long lunch (in the rain), etc, etc.
Umm, cant really think of anything else. The route is terrific, apart from the drag into Ax le Thermes, which is as crappy as any cycling I've done - 20 km uphill drag in the rain with a headwind, on a busy main road.
You'll love it (the rest of it - not the ax le therms bit!