It shouldn't really matter - the player who, on his/her turn, cannot move loses. I have not yet completed a game to that point so I wouldn't know if the final move of the losing player is required.

It is not impossible to win without a queen. I (a very poor chess player) have won at least a few games having lost my queen rather early. Against an experienced opponent, however, that is different.

If there are two separate enclosed sections the only way you can change the outcome is by enclosing part of your section, eliminating some of your moves - as long as both players fill in their section completely, "perfect moves" are every move.

That would assume all the squares are filled, which seems to almost never be the case.