1)
~(よ)うとする means "to try to do" / "to be about to do". ~(よ)うとしている is the ~ている form of it. Therefore, 何をしようのだろう is wrong.
2)
Usually, it's 尋ねなければならない. It can be a conditional, but the omitted part must be "something bad will happen", so what the sentence expresses is almost the same as 尋ねなければならない, anyway.