If water really is coming between the soffit and the facial metal You do have a legitimate leak.
It doesnt have anything to do with not having a gutter.
The one pic showing the shingles are as previosly described, a gouge made while the shingles were hot and also a couple shingles were too beat up to install.
Not wise to do under a window. Lol.
The streaking around the white eve metal at the bottom is totally normal.
It doesnt matter what shingle you use.
I inform all of my custumers before hand
And strongly advice them NOT to use the
White, gray or cream colored eve metal.
Those colors have to be cleaned every year for the house to look nice.
A once or twice yearly duty.
The brown or black evedrip gets dirty also
But it always looks clean from the ground/street.
I can see your dirty, uncleaned white/ light colored eve metal from 100 yards away…
And it usually is the only thing that makes a house look dingy.