How would a professional flash this

I think you're on the right track. I'd apply enough glue so some squeezes out, wipe the face with a damp towel to remove the residue, and either clamp it or wrap it with blue painters tape until dry. Just make sure that the wood remains flat and not bowed when you clamp it. After the joint is dry, remove the tape or clamps, and you should be good to go. If the finish is hazy, polish it with steel wool (0000) or I'd use a rattle can of poly or Deft wood finish to feather it in before you install it. It will take another 10 minutes and better disguise the split. If it's a dark finish, there's "dark wood" Titebond available.

Well I can tell you term for the 1X on the angled portion of the roof, it is rake mold. Most rake molds have a bit more elaborate profile, but there is no rule saying it can't be a simple piece of 1X. The term for the same piece running horizontally might be eave mold, but I have never heard or used it in all my years. This may have been a call made by the installing carpenter, for the idea of keeping things consistent, what is on the rakes, is also at the drip edge.

The blocks are there to keep the gutter from hanging at an angle under the roof edge so the rain does not over shoot the gutter, since it is fastened to the projecting 1X. If it was lowered, to eliminate the blocks, the rain when heavy enough, would shoot over the gutter, only light rains would be caught. If you are using aluminum gutters, you will need fasteners only 24-30 inches apart, unless you have heavy snow loads sliding off the roof, then you may need the fasteners every 16 inches. These blocks would only be needed at each fastener location. The composition roof may not let the snow slide, but the roof has a pretty high pitch.