|Cam the Dolomites in absolutely awesome. So many responses, but not one that answered your questions...
Staying.. As Eduardo said, the huts are an awesome place to stay. Photos of first ascents adorn the walls, they always have a stunning view and you'll be glad to return for a jug of red wine after flogging yourselves all day. Some must stays are the Rifugio Vazzoler (1725m) in the Civetta group. It's one of the most idyllic spots in the whole of the Dolomites to stay.A bit of a walk up a hill, but then you're well placed for an assault on the magical Torre Venezia. Rifugio Auronzo is on the end of the road to Tre Cime, another must visit area. If you can't get in here then the little town of Misurina is very pretty. Rifugio Scoiattoli again is at the end of the road at Cinque Torri, and a primo spot to stay. You must visit the Tofana group and drive to stay at Rif. Dibona. Climb a route on the Tofana di Rozes and then stay up high at the Rif. Giussani before walking back down the next morning. We did this, it was the highlight of our trip. There's a monstous camping/caravan park in Cortina that was pretty cheap too.
Classic routes...so,so many. There's good reason why the likes of Fantini return year after year. The rock is far from choss. The routes are run out though. We looked hard to find the book "Classic Dolomite Climbs" by Anette Kohler and Norbert Memmel. It lists 100 of the highest quality classics between UIAA 3 and 7, soup to about 23. Classics we enjoyed were Yellow Edge on Cima Piccola at Tre Cime, The Cassin route on Piccolossima at Tre Cime, the Tissi route on the south face of Torre Venezia, the Pilastro route up the Tofana di Rozes and the routes Via Miriam and Via Finlandia on Torre Grande at Cinque Torre. The Messner route on the Neunerspitze looks phenomonal too, with the remote Rif. Fanes a great place to stay.
Too many climbers wait for the good conditions over there. They miss out. We got wet most days, but not for long, and we had a ball.
If you can't find the guide book e-mail me and you can borrow mine.