Moderation is the Key
If the dog hasn't has blood work in as recent as 12 months, I would have a complete blood panel done on this dog asap and go from there.
I would immediately stop feeding the salmon, salmon oil and coconut oil. It's not a surprise to me this dog is as sick as you describe. No wonder he is eating grass.
I would treat this dog to a nice roasted chicken meal with white rice and steamed carrots. Give his system a break for awhile. This may just help him turn around.
Feed only white meat of the chicken. Mash the carrots after steamed. Add a little either flax oil or rapeseed based on the amount of carbs and proteins. Wouldn't hurt to add some natural oat bran for fiber, to the cooked meal before feeding.
The dog can do alright on this for the short term but the idea is to give his system a break. The above will be easily digested and will be something much different that the over consumption of salmon and salmon products which I feel is number one reason your dog is sick.
Switching to something like the above can help this dogs body quiet down some.
If you continue to feed whole food for more than a few months, and slightly cooked, you will have to learn how to properly supplement the calcium to balance out the phosphorus in the animal proteins. Number one.
The dog may of had dry skin since 4 months old and obviously it just got worse but I do not feel that the salmon consumption has helped this dog only because it has been way too much over time.
Consider your source of salmon as well.
Treating the symptoms is not treating the cause. Re: Topical solutions can help sooth the dog until the cause is found, treated and eliminated.
From what I understand this dog has been on a strictly salmon diet and that's it?
Has the dog been given Vitamin E supplement during the feeding of the salmon meals? Extremely important. Salmon oil is oxidizing. It needs to be properly supplemented with Vitamin E which is an antioxidant.
Also Salmon is very very rich. Is this cooked salmon? How are you preparing it if it is being cooked? What is included in the salmon meal before it is set down for the dog to consume? How much of everything?
On a rotational whole food diet my dog gets salmon once a week. Sardines on one other day per week. That's it for seafood.
To answer your question about "What other oils could I try on him"?
When, I first seen this thread I knew that E.F.A. would be in play.
When I feed beef, I supplement with Hemp oil. When I feed chicken I supplement with Flax oil. Sometimes I just use rapeseed.
I would eliminate supplementing with coconut oil all together for sure.
Currently I am researching more on the use of E.F.A.'s in a dogs diet and will post my findings upon completion if desired.
All The Best.