There will be alot of snow from the Platforms onward, but in general, the actual rock climbing should be dry and pleasant. A standard axe will be sufficient. Crampons are a personal choice, and for whatever it's worth, I'd probably leave them and expect to be following a bootpack trail whenever snow was encountered. There will certainly be snow descending from the summit to the rappel, and in the gully you have to cross getting to Wall Street. There will also be snow in various places on the upper XM but you should be able to skirt it without too much trouble.
Sleeping high, then getting an early start, summiting, and descending to the valley is a standard formula and should serve you well. Remember that overnight numbers are limited so getting signed out with the Park early is important for securing the site that you want in upper Garnet Canyon. Anywhere from the Caves to the Lower Saddle would do, depending on your party's preferences.
Have a great climb. Despite the very late and snowy spring, my guess is that the mountain will be in reasonable condition and you'll have a good visit.