Having been a few times this would be my take:
6-7" travel bikes with low/slack geometry are ideal, they'll be decent on the BIg tracks and allow you to access more variety of terain (although I manged to get from one end of the Portes Du Sleil in a day on a big hit...)
Personally i'd look at a second hand bike a few months before going and ride that and get used to it (and fix any problems) then sell it when you get home again.
+1 Excellent advice, although if you 4X bike is burly enough you might get away with frame, forks and bigger disc rotors
Morzine is nice and Central, but Chatel arguably has quicker access to some nicer tracks.
Hire bikes are mainly trashed. unless you're willing to pay absolute top dollar. You do get decent support from some of the hire shops, but you might be sat at the shop half your holiday.
Big Brakes are your friend.
Taking a bike with you is a bit of a pain, but knowing that it's in good shape, that it's setup just how you like it, and having the right spares is priceless.
If 5 or 6 of you can find a suitable van to share, and enough time off you might find driving economical, it's going to be tough with fewer people than that. Same with a chalet, self catering is very economical if there are a few of you and you stock up on food and beer before you head into the mountains. A half board Chalet will make it feel more like a holiday, the support and information most reputable companies offer may be useful to a first time visitor.
You'll suffer the fact that there are now less and less Geneva flights in the summer, Liverpool is probably your best bet from Lincs.
Don't forget to factor in the cost of airport transfers and insurance.
You've got to bear in mind that although the total cost can be quite high, it is a holiday and you will get more lift assisted riding in in a week than you probably would in a year in the UK, If you look how much it costs per day to go yo Wales or FT. Bill, Portes Du Soliel might well work out just as cheap.