How to repair or replace a fascia gutter end caps

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.

Ideally you'd disassemble the corner and replace parts so you can start fresh. Sealant over sealant is a low-percentage approach.

Otherwise, it's a matter of scraping away as much of the old sealant as you can, cleaning the surface well (maybe with alcohol), and using one of the purpose-made sealants. It must be extremely flexible to handle movement due to temperature variation.