yes, "I haven't gone camping" is a perfectly used grammar.

I would actually say, not always or often. You can say it. Grammatically, it is ok. But, we would only use it in a situation where you thought about going but didn't. I would say haven't been is more common.

Depends on what period of time you are referring to. If you mean you have not been camping before then you would need to say " I've never been camping before."