Which of the following is correct?
(1) The dog is half wolf.
(2) The dog is a half wolf.
If (1) is correct, is "wolf" an uncountable noun or an adjective?
The correct formulation is: The dog is half wolf.
In this sentence, half wolf forms a predicate nominative which means that the sentence doesn’t change meaning if the subject and predicate are inverted. “Half of the dog’s nature is wolf” is an exactly equivalent statement.
Diagrammatically, the sentence reads “dog is wolf”. The is an article adhering to dog, and half is either an adjective describing wolf, or arguably (and probably better) an adverb that modifies “is” by itself would imply a full equality, whereas the more likely desired result is to say that half is the exact nature of the relationship.
Note that the subject dog could either be the dog (demonstrably a single dog to which your are referring) or a dog, implying that any and or all dogs are half wolf. In either case, however, an article would force the word half to modify wolf, rather than the verb is, and, in doing so, make for an awkward construction. It is awkward because half a wolf is a very concrete (if somewhat messy/gory depending on how you slice it) thing.
Considering that half wolf is another name for the breed Wolfdog (wolf–dog hybrid), the correct sentence should be "(2) The dog is a half wolf."
Wolf-dog Education says Mid-Contents (35%-74%) (are) g)enerally described as being “half wolf”.