In that time period you are highly likely to experience at least one enormous Alpine storm. However, my recollection is that paclite and pro shell are both waterproof, its the breathability and durability that differ, so either will be fine. Many of the French hike in basic ponchos and seem to cope. I use an ultralight PU waterproof which has never been a problem even in high mountain storms.
I'm doing it at the end of June so if you post nearer the time I can let you know the conditions at that point. I'll be in trainers. www.ohm-chamonix.com usually has detailed conditions reports that are also worth checking.
Any snow you do encounter, by mid July, should be relatively limited, well consolidated and easily crossed in trainers. Even in early-mid June I've been fine in trainers.
I do at least some of the TMB every year and all of it some years.
1) re boots. I would echo everyone else here - if you've got some comfy boots, use them. Trainers are fine, it's what I use even early season in snow (sometimes trail shoes, sometimes road). My wife has done some of it in flipflops, tho' I wouldn't advocate that to other people. B1/B2 completely unnecessary. 2 pairs of footwear unnecessary and too much weight.
2) Time of year. Mid July to late August is busy, then you hit the Ultratrail the last weekend of August which means accomodation is rammed. Late June-mid July is relatively quiet, as is Sept although I've been unluck with Sept weather.
Alternatively if you know what you're doing (i.e. don't need to learn how to walk in crampons), you can get a private guide +huts etc. for the 2 days necessary to summit and descend for c. £1k.
You can go solo... but one you've stood on the edge of a crevasse that would swallow a london bus you might turn round.
re resorts that don't need cars: Chamonix is excellent (free public transport with all accomodation), La Berarde is good once you've got there (requires hitching I think). Zermatt easy to get to and good public transport.