The only guarantee for water along that route is Schultz tank which is large and deep and should have good water, however there will be rain (most likely during monsoon season) and you may encounter a dripping, but scummy, spring along the lowest parts of the trail near Weatherford Canyon. It is dry in the Inner Basin unless springs are flowing, but I don't think there are any along the trail, and they will probably have been sucked dry by the City Wells by late July. There is a very high likelihood of water being available at the Inner Basin Entrance in the form of a spigot . There is a tank at Lockett, but I question the water quality there.
The DEX has the accurate mileage, but I give around 9 to 10 miles from Schultz to Lockett Meadow.
I think Monsoon season will begin around June 20, plus or minus 5 days, not by the calendar according to the NWS, but when dew points rise dramatically, and it begins to rain over the Sacramento Mountains. It will start about 10 days later in Arizona.