Against the house you need a kickout flashing. The rot could also be from water getting under the fascia, not sure how the roof butts there but the little 1" or 1.5" flashing looks to be on the face of it rather than under the fascia.
The open end gutter can always be capped and you can install a downspout as well. The rake "drip edge" looks to be blocking the flow and maybe turbulence caused the water to get behind the fascia somewhere before.
The open end gutter can always be capped and you can install a downspout as well. The rake "drip edge" looks to be blocking the flow and maybe turbulence caused the water to get behind the fascia somewhere before.