The weather is a lottery. It is currently very hot where we are. The Air Temp may only be getting up to mid to high teens but in the sun it is scorching. It is shorts and T Shirt weather here at the moment (we are at 1200m and I'm sun burnt already) and perfect biking weather. The mountains are already echoing to the sound of motorcycles.
At some point soon "The Ice Man" will return for a few days with some very wintery weather just to remind us winter has not gone to far away.
June can be a bit wet at times so you can expect a few days of rain. It is a bit of a thunderstorm month in the Alps and across Bavaria. Be flexible with your trip and you can usually find some better weather. The Grossglockner and Nockalmstrasse are scheduled to open on 1st May. Stelvio, Timmelsjoch etc usually open at the very end of May, beginning of June. If the East Ramp of Stelvio is closed, check out The Umbrai Pass as often that is open and will take you up to just short of the top of the West Ramp. Gavia usually mid June. Most of the great roads lower down aren't troubled at this time of year. The likes of Gerlos, Rossfeld, Postalm etc are open all year and only fresh snow fall will trouble a motorcycle. It doesn't hang around long on the asphalt this time of year.
Accommodation isn't really a problem. Some places will be shut but many will be open and there are far less people around for the spaces available.
We are open almost all year round by the way