Maybe I can answer some of your questions and add my 2.5-cents.
1.does anyone know what the name of this system is called?
A. It is a simple, stick built, triangular roof system. Normally the roof system consists of a ridge board, rafters, and ceiling joists.
Discussion: Reportedly a ridge board is not present. (I can't tell from the photos.) Presume the rafters are secured to each other in some manner where they **** at the peak. The 23/32nds" roofing (when well nailed) will prevent the rafters from deflecting. The bottom horizontal member, referred to as a "ceiling joist" is, from what I was taught, a "span". What's the difference? A ceiling joist is intended to support a ceiling, a span is intended to keep the opposite walls parallel. Usually ceiling joists are on 16 or 24" centers, spans of 48" centers or more are common. The vertical boards serve as a support to the spans and is provided to keep the span horizontal and keep it from sagging. Ideally the span would be one piece. Of concern is the fact it consists of three pieces that are not adequately connected or joined.
Corrective measures could include the sistering a 2x12 x width of garage as has been suggested. Additionally, install blocking between spans/ceiling joists. Make certain the walls are parallel when working on the spans. The rafters could be reinforced near the peak by nailing a 2x6 on the inside of the rafters parallel to the ridge, for the full length/width of the garage IF there is any concern about them.
2.can I safely insulate the rafters?
A. Yes after the proposed modifications are completed.
3.can I safely insulate the ceiling joists?
A. Yes, after the proposed modifications are completed.
4.can I safely add a drywall ceiling or drop ceiling?
A. Yes, after the proposed modifications are completed.
5.can I safely add any storage?
A. Good question! This is where the span tables need to be consulted. My recent build has No. 1 2x12's on 12" centers for a 22' span specified. It also has a vertical 2x4 support located about six feet from the outside wall supporting the rafters. With all of that lumber the floor loading is still limited per the engineers drawing.
6.any other thoughts?
A. Yes, I wish you good luck! And, the customary disclaimers apply.