May and June are peak seasons. Everything will be expensive. Make sure you prebook everything as on spot prices will be quite expensive. Snow can be found in Gurudongmar, zero point an probably Nathula
Yes if you want to get away from the scorching heat of the sun.
Secondly, remember, if you want to save some money book everything you can in advance beacuse everything will be very expensive.
If you do go there have fun.