To respond to various above posts, every member there is structural. The loads placed on any particular member vary in direction and magnitude. Any member that is removed, needs to have it's loads transferred to either new members, or an existing member that has been beefed up to take the additional load, unless that member is already adequate to carry the additional load.
This post should be in all caps, but I will assume the OP can read, understand and follow it without the emphasis.
The ridge board/beam is a 2x, rather than a 1x, so it does act as a beam more than a board, for the short span there. And the vertical 2x4's transfer a portion of the roof load to the strongback, which in turn, transfers the load to the ceiling joists. In addition, the collar ties contribute to resisting the tendency of the rafters to spread.
In comparison to many garages that I have seen (Including many hip roofed ones), this garage has the requisite number and spacing and size of ceiling joist needed.
The strongback shown is, I suspect, NOT made up of a bunch of small pieces as shown by the red lines in the photo, but rather, made of two or three longer pieces. The OP can verify with additional photos from the other side. This strongback does a few things that are important. In addition to transferring some of the roof load to the ceiling joists, it also helps with resisting side forces on the ceiling joists and tying the end walls together as mentioned above. It is a key element in the structure.
The drawing above, by Chris705, goes a long way toward addressing how to modify the existing structure to accommodate this lift. One thing that isn't shown is what happens over the lifted portion of the ceiling. Where the two existing joists are cut out, they could be replaced, but 1 1/2 feet higher, and secured to the rafters, to tie them together.
Another thing to check is the roof slope, to see if there is adequate headroom at the hip end, for this tray ceiling.