In reply to B&H:
Your post needs clarifying. Z > C is against the flow - but there's nothing at all wrong with that, but it does have disadvantages. In the other direction it's usual to do it in about six days. Fourteen days is very generous.
You mention peaks - which peaks have you in mind?
You'll have to define 'classic' as there are numerous ways.
Zermatt IS the start (or finish). Your first hut would therefore be Sch£el. Very tricky to get down onto the glacier and then up on to the Stockji from there so needs checking the night before.
The Chanrion > Valsorey leg is also going to be a bit tricky in that direction - getting down from the Plateau du Couloir towards the hut late in the day risks being exposed to stonefall - especially if July has been hot. Even in the other direction it can be dangerous - even in the night/early morning if it doesn't freeze.
The glaciers are generally OK in July, it's later on that they get drier (usually).
The Haute Route by Peter Cliff is still about your best bet.
Yes, it's feasible to organise by yourselves. Needs a bit of research, that's all. Be sure to book huts. Don't just turn up.
Post edited at 17:59