any tips on what I should be looking for with appropriate or faulty installation?
To answer your question better after my initial shock, it won't be what you surely want to hear.
All of it is installed wrong, not just in small workmanship details, but in how it is done overall with wrong methods and wrong parts (or lacking parts thereof).
- The membrane doesn't look adhered properly, and I would also question how the substrate (the plywood/insulation board under the membrane) was done/replaced, and if it slopes where you want it to go. Can't tell from the nightime picture if that is the plywood popping up.
- The termination at the 2nd floor wall is wrong and with no counterflashing. At the corners of the 2nd floor it will require some flashing shapes as well.
- You don't wrap the edges of a roof, you stop it at a drip edge. Doing the edge detail properly will naturally eliminate all those "xmas present wrapped ends" that are bulbous, folded, and air-pocketed.
- Then install a flat board or metal covered fascia, and the gutter attaches through this fascia, not the membrane. The overhang may need some detailing/blocking depending on what is hidden behind the membrane.
With flat roofs, you should hire a licensed roofer who is certified for that particular product. They will know how to do it right and will warranty it. Note that they will not reuse any of the materials that are there now, so all is lost. But better than continually chasing all the rot and leaks that you will get with this current installation.
With the gutter work and interior millwork supporting his ineptitude, it seems as though this person took advantage of you and sold you a bill of expertise that doesn't exist. I am afraid to know how much he took you for. He should not be let back on the property at all. If you are elderly, there is some additional recourse against financial exploitation and consumer fraud. Just don't be embarrassed by it, it happens to people and many have asked advice about contracts/relationships with contractors on this forum.