The cableway from Weggis to Rigi Kaltbad, which could be used as part of one answer to #2, will be closed for maintenance during that part of May.
1.- 0915. The top station is "Kulm" not Klum.
2.-Yes. There are several possibilities, which can be understood by using the SBB timetable, which also shows the schedules for the lake boats and the cableways used for public transportation:
This link also is very helpful:
The cograil to Pilatus Kulm from Alpnachstad is operating on those days in May. (Answer to #3.)
You can come down from Pilatus to Alpnachstad by cograil, or to Kriens via cableway.
From Kriens, go to Luzern by bus.
Then boat to Vitznau, or train to Arth Goldau.
Or, from Alpnachstad, go by train or lake boat to Luzern, then onwards as noted.
Once in either Arth-Goldau or Vitznau, take the cograil to Rigi Kulm.
Those journeys take roughly 3 hours.
Also, this website gives the details of the lakeboat routes and schedules:
Scroll down for the correct seasonal time table. Use the Spring Timetable.
In May, it is not possible to go by boat(s) directly to Vitznau from Alpnachstad.
It appears that you can only take boats from Alpnachstad, via Kehrsiten-Bürgenstock , change, and continue by boat to Vitznau on the Summer schedule, after July 31. Then, there are two boats at 0914 and 1017 from Kehrsiten-Bürgenstock to Vitznau. The one at 0914 runs daily from August 1, the one at 1017 runs on Sundays and Holidays from August 1.