Your examples 1 and 3 are almost idiomatic. The hyphen is not necessary in this case--why not is complicated and I won't clutter up this answer with it here.
Your examples 2 and 4 are not very idiomatic as written. It would, however, be perfectly idiomatic to restate number 4 either as:
I like to go mushroom picking in the summer
I like to pick mushrooms in the summer
The situation is different when there is a preposition in the mix. So, for example, it would be idiomatic to say:
I like to go foraging for mushrooms in the summer
but "go for mushrooms foraging" does not work in this context.