An increasing number of huts in Austria / Italy (less in Switzerland) has showers, but still not the majority.
Huts in Austria and Italy normally have some 2-bed-rooms - but they are the first to be occupied.
Camping near the hut and using the facilities of the hut depends on permission of the warden. If you ask polite and offer some money I guess it will be accepted. Eating in the hut is no problem.
Weather in the alps isn't very stable. Afternoon and night showers are common even if the weather in general is fine. Take this into consideration for camping.