Why are the new roof shingles showing these lighter patches in cool weather

A skylight (and any hole in the roof) needs flashing, not caulk. The flashing should be overlapped so that shingles above are on top of the flashing, and as it goes down the side, each piece is on top of the piece below, so that any water runs down the roof rather than in your home. The flashing should go up the side and under any lip from the top of the skylight so that rain from above cannot get in behind the flashing. And then all nails should be covered in a roofing caulk and under a shingle/above flashing/other protective layer.

Replace it now. Like TODAY. That's BAD.

With that kind of bad shingle wear, your shingles are likely to create ice dams which will cause water to back up and leak through the roof, causing interior damage, compounding the cost.

In the summer, the roof won't hold the water because it can't freeze, so the curls will only grab a very little water before it drips free.

In the winter, however, a little retained water will freeze, expand, and capture more water, which will freeze, and so forth.