Let' start with some questions, if you don't mind.  

How long has the tree been in the ground?

How well does the soil drain?

Have you had a soil test recently?  King Co. offers free soil testing.

How do you fertilize the tree?

How do you water it in the summer?

Which direction is the affected side of the tree facing?  South?

Has anyone--you or a neighbor--used an herbicide this spring?

Do you have any photos of the leaf spots on your tree?

Did you notice any spots on the apples this tree produced?

Does the tree seem to leaf out, then the new leaves turn brown and fall off?

Do you see any discrete areas of "stringy" bark that kind of span an affected area like the strings of a guitar?

Photos:

Can you get me in-focus photos of the affected areas you've sent me?  I would like to see what you mean by the orange sticky stuff. 

I would also like to see closeups of the black areas on the branches.

Can you send me a photo of the entire tree?

Jen--I can't access the folder you shared.

Okay--I've compiled the photos you sent the second time into a gallery shot--they are numbered.

The easiest first--# 11 is burrknot: https://pnwhandbooks.org/plantdisease/host-disease/apple-malus-spp-burrknot  These bumps are the beginnings of new roots on the branch--some rootstocks are more susceptible than others.  If there are only a few, you can remove them with a knife or a blowtorch.  I think I'm seeing evidence of burrknot growth at the junction of the branches in #8, too.  This looks like a very unstable union that could break easily.  The problem with the burrknot formation is that it opens the tree to insect damage and disease.

I'm actually not positive about the second, more serious problem (photos 2, 3, 8, 9, 10). I was considering Apple Anthracnose (see below) or Black Rot (Diplodia Canker), and I think the symptoms your tree is showing look more like Black Rot--however, it's more common on the east side of the Cascades. The fungus enters through damaged tissue--such as the burrknots.  Excerpt from PNW Handbooks:

"Symptom: The bark of infected limbs becomes slightly sunken with reddish-brown areas. The canker may just be a superficial roughening of the bark or develop into necrotic cracks with discolored vascular cambium. Cankers can become a few feet long.  Leaf symptoms include the development of a leaf spot with a purple margin and tan center commonly referred to as frogeye leafspot. Infected leaves fall prematurely from the tree."

Unfortunately, the only treatment is as follows: "Remove and destroy dead wood, wood prunings, and mummified fruit."

If this is indeed Diplodia Canker, it's the first time I've seen it.  Anthracnose is more common, but those tan leaf spots with the dark outlines are so distinctive.  This article from UMN explains it well: https://extension.umn.edu/plant-diseases/black-rot-apple

My second choice is Apple anthracnose (ann THRAK nose), but again--the symptoms aren't as close a match.Pictures 9 and 10. http://hortsense.cahnrs.wsu.edu/Search/MainMenuWithFactSheet.aspx?CategoryId=3&PlantDefId=59&ProblemId=1 The photo in this article doesn't show the early stages of the disease. You can see this better here and here.  

Personally, I would take the tree out, and replace them disease resistant varieties.  You can find more information about that here: Growing Tree Fruits and Nuts in the Home Orchard https://catalog.extension.oregonstate.edu/sites/catalog/files/project/pdf/ec819.pdf  None of the possible diagnoses have any good treatments, especially on a tree that has so much vulnerability to disease and insects due to the cankers and burrknots.

I am going to send your photos to the WSU Plant Pathology Lab for a second opinion.  I will get back to you when I hear back from the plant pathologist, Jenny Glass.  This is her busiest time of year, so I don't know how long it will take. Please don't respond to this until I hear from her because it "re-opens" the question and she won't be able to access it then.  Thanks!

Hi, Jen--thanks for your patience.  Jenny Glass, our WSU Plant Pathologist extraordinaire got back to me this afternoon.  She thinks it's probably anthracnose because it's a common apple disease in our area, and she has not seen Diplodia Canker here before. Follow the instructions give in the previous response for anthracnose management. You may also see bullseye lesions on your apples, too, as anthracnose doesn't only affect the woody parts of the plant. It also looks like you have some classic powdery mildew on some of the leaves--see image 13.

Here are some of her comments:

Prune off the damaged limbs well below the affected areas.  You should also remove the plant tags--they could girdle the limbs in future years. 

"Spots in picture 2-3- I see this from time to time on apples. Usually can’t pinpoint a specific pathogen so have started to wonder if it might not just be a plant defense response to a variety of issues. Not a big deal- plant makes abundant green tissue in excess of its needs. Here is something I wrote recently: The distinct brown spots are likely something else but not something that is well understood - see https://pnwhandbooks.org/plantdisease/host-disease/apple-malus-spp-necrotic-leaf-spot (so perhaps a disease or very successful ‘hypersensitive reaction’ by the plant as a way of defending against parasitic pathogens). I frequently receive samples with these symptoms in apples submitted to the lab but never have much further to report after incubation/culturing tissue onto sterile laboratory agar."

Here are some resources for managing your problems.  I think this will come in very handy as many of the problems you are seeing this summer need year round or other-season management: 

WSU Hortsense--apple problems.  Look at the list on the left, and refer to Anthracnose, burrknot, and powdery mildew.http://hortsense.cahnrs.wsu.edu/Search/MainMenuWithFactSheet.aspx?CategoryId=3&PlantDefId=59

A couple of really terrific resources:

Managing Diseases and Insects in Home Orchards https://catalog.extension.oregonstate.edu/sites/catalog/files/project/pdf/ec631.pdf

Growing Tree Fruits and Nuts in the Home Orchard https://catalog.extension.oregonstate.edu/sites/catalog/files/project/pdf/ec819.pdf

The very best of luck to you!