Hey MM. A basic Air Force turbine airstart cart is the A/M32A-60, which was used in Viet Nam and a variation is still in use today. In the navy we call called them "huffers". For flight deck use, huffers were usually found attached to the back end of a standard MD-3 shipboard tow tractor. We also had huffers similar to the Air Force units that were towed and not attached to a tractor.
I can't address F-105s, but, in the Navy, we used a variation of a bottle jack to change tires on F-14s and F-4s.
I've attached a picture I pulled from the web of an A/M32A-60 start cart and an F-4 in Viet Nam.
Hope this gives you something to work with.
