... so if I understand correctly, by volountary deciding to leave this card in the staging area, you are deciding to make the whole scenario a Siege Quest one? Is that correct?

Yes. The quest will remain a Siege quest so long as that location is in play, so if you just leave it in the staging area and never explore it then that's the entire quest on Siege.

So, even playing the Trained for War event will not change it to a Battle quest, because The Cross-Roads give the current quest the Siege "status"... correct?

Correct. Trained for War received errata so that it can't override Siege:

"Play only if each hero you control has the printed Tactics resource icon. Action: Until the end of the phase, if the current quest does not have Siege, it gains Battle."

Correct. Trained for War received errata so that it can't override Siege:

"Play only if each hero you control has the printed Tactics resource icon. Action: Until the end of the phase, if the current quest does not have Siege, it gains Battle."

... means that I have the errated version. My real question was : does player cards "overide" encounter cards, when both change something on the board in different ways? But it does not matter. The Cross-Roads changes the status of the Quest but it's mere presence, so as long as the card is in play, this status is changed.