Should I expect any rain on the mountain? I know there will be lots of snow and stormy weather, but does it (ever) get buckets of rain instead?

Lower elevations of the Kahiltna Glacier can get rain in June. However, the rain often tends to be more of drizzle, rather than a downpour.

I was on the south side of Denali one year right around June 1st-June 10th.

We had daily torrential rain up to about 10,000 feet. It made glacier travel rather difficult and skis were required (otherwise it was post hole city). Everything was always wet. Eventually we switched to climbing at night to avoid being out in the rain. The conditions on Reality Ridge up to to 10,000 feet were appalling. Really soft, rotten snow, no belays.

I like May a lot better.