Attaching gymastic rings / pull-up bar to the ceiling

1. I would recommend that you don't use a washer on the outside of the Unistrut bottom that has any chance of wedging the Unistrut open. The fender washer on the inside is definitely the simplest solution, and it has the advantage of leaving all of the unistrut opening clear for your platform hangers.
2. The strength of your lag-bolt connection is based on the surface area of the threads. Therefore the length of the bolt is potentially just as important, if not more so, than the diameter. Remember, you have 1/2 or 5/8 inches of drywall that isn't giving you any grip, so your 2 inch lag bolt is only holding on by 1-1/2 inch or less. Your joists will be likely 8 to 10 inches tall. 5/16 x 4 inch lag bolts will give you more holding power than 3/8 x 2.

2x's laying flat as opposed to on edge are going to bend or break when weight is suspended from it over time. The longer the board the faster it will compress or fail. To remedy this problem you must add support to the existing framing members. I think the simplest way is to install additional 2 x 4's (on edge) attached against the edge of the original 2 x 4's. It should have a "L" shape profile. You didn't mention the span of the rafters, but you may consider adding webbing above the platform. All this is are lengths of 2 x 4 attached to the platform joists at one end and the other to the mating roof rafter above. This is probably over kill, but still another way to prevent a failure.