They are called mosquito sanctuaries. ;-)

I wish I could help; I would call a gutter company and they can determine what you need.

Commonly called a built-in gutter. You might seek out a metal roofer that does commercial work.

We usually call it an integrated gutter down here, but you get the idea.

Easy to repair, find someone certified in hydrostop- they will remove the bottom couple of courses of shingles, then they will install the hydrostop system over the whole gutter area and up under the shingles part way.

We find this is the prefect time to install larger scuppers and gutter stops as well.

Rolled roofing is the budget option but results in lots of seams and reduces the size of the gutter reducing its efficiency.