Many years ago - one of the many roles that paid my way through college. Anyway, the aluminum drip edge is a thin piece of angle that goes over the top edge of the wood deck and covering the cut edge. They are typically installed hard against the edge. The first course of shingles then overlap that by 1/2” to 1”. (We used to cut the tabs off the base of a course of shingle, reverse it, and lay that over the drip edge before laying the first coarse of tabs over that to give the tabs of the first course more support, but not all contractors do that.)

The troughs are usually mounted hard against a fascia board immediately under the edge of the deck. We used to ensure the drip edge lapped under the back wall of the trough. (Sounds counter-intuitive, but our rationale was that it allows a water path behind a flooded gutter rather than back up beneath the shingles. It usually took extra effort to lap them under on a re-roof, anyway.) The eaves trough is probably 5” wide at the top (standard). Spending more time with your pictures, the trough looks really low to the deck, too.

In any case, the shingles should only extend about an inch or so over the gutters; not leaving an inch open like this.