Hi there,
It's nice to see your concern for animals.
It seems like you have three concerns here, but please correct me if I am mistaken.
First, you worry about diseases/parasites that may infect your dogs.
Second, you worry about the chickens being properly cared for.
Third, you worry about rats and the ugly blue tarp you see from your yard.
I can assure you that your dogs will not catch any diseases or parasites from your neighbor's chickens. For instance, bird lice don't do dogs or people.
Chickens do not require a whole heck of a lot of daily care. If they have a constant supply of food and clean water and a shelter they should do fine. If you believe the shelter is not adequate you may want to post a photo of it here and ask that question specifically.
Chickens do not attract rats. Open bags of chicken feed may attract rats, but a maintained chicken feeder in a coop is not a serious problem. If it does attract rats, you already have a rat problem.
The ugly blue tarp is an entirely different question. I would be very upset about the tarp, and I would ask the neighbor to replace it with something less hideous. When my neighbor had an ugly blue tarp on his coop (which shredded in the wind and left my garden festooned with blue plastic icicles) I offered to help him replace the roof of the coop with new plywood and roofing material. It took a year, but he got it covered this year before the rains. Sometimes it is cheaper, easier, and more effective to offer help than to complain. Maybe you will get some fresh eggs in return!