A. No change <-- This one!
B. The dogs coat was matted, yet he looked very clean.
We need an apostrophe to show the coat belongs to the dog.
C. The dogs' coat was matted, yet he looked very clean.
The apostrophe after the s in dogs is saying that there are multiple dogs, but the sentence is saying "yet he looked", so "he" means there is one dog.
D. The dog's coat was matted; yet he looked very clean.
Semicolons are used to link two independent clauses that are related to each other, but "yet he looked very clean" is not an independent clause, so we just need a comma.