1.Are the cards under the Crossroads part of the encounter deck?

2.In stage three of Helm's Deep you add Helm's Gate in the staging area before making it the active location.I may use Thror's Key at that moment?

3.Beginig of the quest phase is before commiting characters?

1 Cards underneath the Crossroads is not part of the encounter deck

2 You can use Thror´s Key at any location, which is not immune, in response to it entering the staging area. So in a word: yes

3 Yes beginning of quest phase is before anything else occurs, just as after a phase is after everything else has occured