The Spanish (Armaris) Strategic Projection Ship, which is one of the 2 competing designs for Australia's Amphibious Warfare (LPD) project is designed from inception to operate STOVL fighters, such as Harrier or F-35B as WELL as helicopters, vehicles, troops, medical facilities etc. As such it possesses both the space and facilities to operate both, including the necessary spare parts, ordnance etc.
This vessel also comes standard with a "ski-jump". Australia would have to "modify" the vessel NOT to have one. A ridiculous idea in MY opinion. Even if we don't operate STOVL aircraft ourselves, it would be useful to carry or assist other nations that do, if necessary, even only for "emergencies" or for short durations.
Details on Amaris's website show that 6 aircraft can operate from the flight deck and 12 aircraft in the hangar. If the light vehicle deck is also used up to 30 aircraft (including 20 JSF sized aircraft) could be carried, meaning that 10 CH-47 Chinook sized aircraft could still be carried IN THE HANGAR, even if the total load of aircraft would be carried.
Under these circumstances, it would a relatively simple matter for one of these vessels to carry 12-24 F-35 aircraft, and still carry up to 12 helo's, albeit with restrictions to the light vehicle carrying capacity, depending on how many vehicles you wished to carry.
These vessels also operate a tank deck, and allows the carriage of up to 41 main battle tanks or self propelled artillery vehicles.
Depending on mission requirements, one or both of these vessels could carry F-35's and STILL carry a significant amount of helo's, vehicles and landing craft. Irrespective of how many aircraft/vehicles/vessels are carried, each ship can carry over 1000 troops and has medical facilities with over 50 beds, plus ICU's, surgical/dental wards etc.
The French Mistral vessel, the other competing design, has far more limited capabilities, on paper. It can only carry 450 troops in "permanent" accomodation, though more can be carried for short durations (up to 900, I believe) no STOVL aircraft can be operated, and it has far less vehicle capacity.
This competition is a no-brainer in my opinion.
As to the F-35B question, it hasn't been decided yet. RAAF are still evaluating the options. If a small quantity of F-35B's are acquired, they will be operated by RAAF. The fleet air arm has long given up it's fixed wing component and I serioulsy doubt we'll ever see it back.
I think RAN would be so thrilled if this capability were to be sought, that it wouldn't care about the RAAF operating off it's vessels...