The sentence is okay but needed is not usually placed before the noun, even though it is an adjective. Normally we would say 'the one (which is) needed' Equipment is singular but refers to more than one thing, so it would be better to say '...in order to bring only what is needed.'

Yes, it is correct. He wants to bring the equipment that is needed.

Necessary is a little deep, but you can say either, inorder to bring only what is needed as teacher jacquid said or inorder to bring what is necessary. Either one is understandable and correct.