Hey - GOOOOD questions, well done!!
Number 2 and 3 are both regular and accepted ways of asking this question. It's all about the emphasis when you actually ask these questions.
Question 2, for example: "Are you going to RUN as WELL as hike?"
Question 3: "Are you going to both run AND hike?"
Number one is different for 2 reasons:
1 - "moutain-climbing" is an entirely different sport to hiking.
2 - It would be.... "as well as "mountain climb" in the non-continuous form, like you have done with hike.
Any further questions?