Is there a name for adjectives that take the form of "noun-verbing", like "rabbit-hunting" or "self-driving"? Do this form only occurs in English?
Since in your examples the 1st component determines the 2nd one (not just hunting, but specifically rabbit-hunting, not simply driving, but self-driving), such compounds are of the tatpuruṣa (aka tatpurusha) type. It is a term of the Sanskrit classical grammar, in the European tradition such compounds are called endocentric or determinative, where the compound is essentially the sum of its parts, the meaning being an extension of one of the parts. Endocentric (lit. “with center inside”) are opposed to exocentric compounds (lit. “center outside”) in which the resulting meaning lies outside the components, e.g. redhead is not a kind of head and pickpocket is not a pocket, those are kinds of people, while in the endocentric compounds rabbit-hunting is a kind of hunting and self-driving is a kind of driving.
This classification is based on the inner structure of the compounds and relationship between their components. In English, apart from difficulties in classifying ing-words into nouns, adjectives, gerunds, and participles, there is also a tradition of writing the parts of compounds as separate words without even a hyphen, so it looks like a good strategy to classify the structure of constructions rather than classifying compound adjective separately from compound nouns or verbs.
Italian has something similar, but it's not "noun-verbing" as much as it is a "verb-noun" object. The verb part is the second-person singular imperative form. Examples:
- A towel is an asciugamani, literally a "dry-hands" or a hand dryer.
- A handheld garlic press is a spremiaglio, literally a "squeeze-garlic" or a garlic squeezer.
- ... or schiacciaglo, literally a "smash-garlic" or a garlic smasher.
- A cassette tape player is a mangianastro, literally an "eat-tape" or a tape eater -- a pejorative term that I sure was well-deserved.
One to add, that I looked up to make sure I had it right:
- A lifebuoy is a salvagente, literally a "save-people" or a people saver.
But sadly, a rabbit hunter is just a cacciatore di conigli.
This type of language exists in other languages; the example I know of this occurring is in Japanese, where you can have a noun and verb follow each other in order to describe another noun; however, you need that second noun to exist (or, alternatively, the second noun can be omitted if context regarding the noun is given elsewhere in the conversation). If it's not there, you would just be describing something normally (ie. it would be turned into "I'm hunting a rabbit" instead of "rabbit-hunting XYZ").
To use your example of rabbit-hunting: うさぎを狩る人, where
- うさぎ is rabbit
- を is a particle (which is often ignored in daily conversation and, for these purposes, can also be ignored here)
- 狩る is the verb for hunting
- 人 is the noun that is being described
And when translating, is more literally read as "rabbit-hunting person".