Hello Sasha,
I just got back from a Mount Whitney zone trip, one week ago. Above 12,000 feet, it is ice axe and crampon weather. With lots of skill, it is not less than ice axe weather.
I wore one layer of goretex, polyester/nylon pants and shirt, and thin polyesther long underwear, at most. Thin gloves, wooley knit hat and mild snow boots rounded out my clothese. These were entirely adequate.
If you wait till after the beginning of June, you will barely need an ice axe.
Water is flowing from Trail Camp, and below. Consultation lake is 99% surface frozen, as is Iceberg lake, and Guitar Lake.
I don't believe it even quite froze at night at 12,000 feet, and the weather was extremely fine the entire time.
Exposure at the top of the 99 switchbacks (on traverse) was the scariest moment of the whole trip.
CHEERS