Beginner Gardening: Palm Identification, Problems & Rose by Any Other Spot, 0 by OnlyThis

OnlyThis wrote:
My husband and I recently rented a house in the Pacific Northwest with a fabulous back yard. Unfortunately, several of the plants in the yard are sickly, and being somewhat new to outdoor gardening, I find myself at a loss. Google searches have done nothing to solve the issues, and I was hoping y'all might offer some suggestions and help to a newbie.

First, there is a palm tree/bush I'm having a heck of a time identifying. A photo of it is in the attached photo montage. What is this tree/bush?
Second, whatever this tree/bush is, it has some major black spots all over the lower leaves (you can see them in the photo). Actually, these used to be worse. I've trimmed off quite a few of the really bad ones. Any advice on how to combat these spots?

Roses - there are several rose bushes that have spidery brown... blobs, I guess, all over the leaves, and a lot of the new buds wither and turn brown before they even bloom. I thought it might be black spot, but the spots don't look right. Is it black spot? Something else? How to combat this?

Finally - this sad little tree in the yard. Again, refer to the photo in the montage I've attached. It's covered in lichen, which I understand is harmless and only preys on trees that are already weakened. Only the top half of the tree has leaves, and the trunk is warped. Is there any way to save this tree?

Thanks for any ideas or suggestions you guys can offer!