I wish he'd get/have a haircut.
This sentence is not a causative construction in the strict sense of that expression since there would have to be an additional verb after "get/have." It is exactly comparable to "I wish he'd get/have a massage/the operation/a heart transplant."
I wish he'd get/have hair cut. (??)
This sentence does use an English causative structure and is fully grammatical; however, it is extremely odd and almost certainly does not convey the meaning intended. The rules of English usage normally require a personal pronoun to be used with body parts associated with a known body, unless you are referring to a mass noun and there is a contrast between all of the mass in the body and part of it (E.g., "I am having blood drawn" versus "I am having my blood tested")
In the case of a haircut, the presumption is that all the hairs on the top of the head will be cut, and so you must say: "I wish he'd get/have his hair cut. You could, however, say: "I am just having hair cut on the sides." In this case, the entire mass of hair is not affected.
Have your hair cut.
This sentence uses the causative construction, since "cut" is the second verb, and is perfectly normal. It could be used in a variety of situations.
Have your haircut.
This sentence does not use the causative construction and is grammatical; however, it is marked and appropriate only for some purposes. It is directly comparable to "have your operation/your massage/your fun/your lesson." It presumes that whether or not the event should take place is in doubt or disputed and affirms that it should proceed despite this doubt or hesitation. It would normally be said with sentence stress on "have."
Get a/your hair cut.
This sentence is normal for American English and almost equivalent to "have your hair cut," despite the difference in structure. I think the difference is that "get" also retains the meaning of "obtain," which is appropriate for this circumstance. In other words, "get" can be used semantically to refer to a result that can also be expressed using "have" (or "get," as well) in a true causative construction.
Get hair cut.(??)
"Get hair cut" would be extremely unusual without the article or a possessive pronoun and would strongly focus on some generic contrast between "hair" and something else. Even in such a case, it would be normal to use the possessive pronoun.