Q) “What type/kind of (a) compressor is used in a rocket engine?”
A) “The compressor used in a rocket engine is called the rocket-engine compressor and it is not that type/kind of (a) compressor that is commonly used in other fields of engineering……..”
All that I’m confused of is whether or not to use an indefinite article before “compressor” in the above two structures.
And would it be grammatical, if I pluralized the above interrogative structure as below?
Q) “What types/kinds of compressors are used in rocket engines?”
There is a strong tendency that indefinite articles are not used even if they are mandatory, so we can see “what kind of person / fool” without “a” quite often. I think it should be regarded as a trend instead of any grammatical rule change.
I would suggest using plural nouns after “of”, i.e., what kind of compressors are used in a rocket engine? There is a possibility that more than one compressor might be used in a rocket engine.
Regarding your Q No. 3, it is perfectly grammatical if you are inquiring about name of plural compressors knowing that there are different type.
A type of A compressor, B type of B compressor, C…
But, there seems to be no difference between “what kinds of compressors” and “what kind of compressors” as there is not a big chance that answers could be different depending on which of the two questions being asked.