Flat roof question/shingles

I would like to ask a question related to shingles, roof pitch and options. My neighbor has a house with mulitple levels. Some roof sections are shingled, some are flat. The problem is that while the flat portions are covered in what appears to be a black tar paper of sorts, it does not appear to be well-sealed or extended to promote water flow into the gutters. In some cases, this material is practically turned up diverting the water from the gutters over the side and onto the flashing. In others, water is "gripping" the edge and flowing between the gutter/fascia.

I understand that shingles require a certain pitch, so he cannot simply shingle over the rest of the area like the steep roof section.

Here are a few pictures:

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The first pic shows the "upper" level right corner of the roof and the gutter. You cannot see the water in the pic, but you can see that the roofing material bends up just as it reaches the gutter and directs the water over the side edge.

The second pic shows a similar right-edge picture of the lower level addition. It was misting when we took this picture, so you can see the rain from before has collected on the edge and is not flowing into the gutter below the edge. Again, instead, it is flowing over the right edge onto the flashing.

The third pic, although a bit blurry, is from just in front of the upper, unshingled roof and shows the left side of the roof that is shingled. There is no problem with water not flowing properly into this gutter. It should be noted that all the gutters and roof are relatively new (gutters recently installed, roof was redone a few years ago and is in good shape).

The fourth shows the same corner as in pic 1 from a little higher perspective. It also shows some spots where the roofing material appears to need some tar applied to seal edges and you can see little troughs along the front edge similar to the lower-level shot where water is sitting rather than draining.

Due to the relatively low pitch, shingling appears to be out of the question. I was thinking that we could re-apply tar or something to sure the seams, but wanted to ask whether there were other suggestions. Would a bead of caulk along the right side direct the water toward the gutter and be sufficient to at least keep the water moving forward? Is the standing water detrimental to the health of the roof (sitting on this paper) or is it not-ideal, but not something to require a great deal of concern?

Any suggestions are appreciated.

Thanks. Dave