Say I played Mimikyu V and used its "Dummy Doll" Ability on my turn. Then on my opponent's next turn, they play a Path to the Peak stadium. Does that cancel the effect of "Dummy Doll"?
Since the Dummy Doll Ability activated and placed an effect before Path to the Peak came into play, its effect remains.
Source: TPCi Rules Team (2021-07-01)
If my Mimikyu V with the "Dummy Doll" Ability protection is moved to or from my Bench during my opponent's turn by a switching action such as Boss's Orders is it no longer protected?
It is still protected. It is an effect from an Ability, not from an attack, and so is not removed by moving to or from the bench.
Source: Battle Styles FAQ; TPCi Rules Team (2021-03-18)
If I play Mimikyu V with the "Dummy Doll" Ability to my Bench and then make it Active, does it keep the protective effect of "Dummy Doll", or is that removed when it moves from the Bench to the Active Spot?
You keep the protective effect of the Ability; it's not like losing an attack effect when moving from the Active Spot to the Bench.
Source: Battle Styles FAQ; TPCi Rules Team (2021-03-18)