Yes, a Digimon with the names gained from the [All Turns] effect is considered to be moved. For example, if this card with a level 3 [Agumon] in digivolution cards moves from the breeding area to the battle area, a Digimon with the names [Greymon] and [Agumon] is considered to be moved.

Yes, it's considered to digivolve into a Digimon card with the names gained from this card's [All Turns] effect. For example, if a level 3 [Agumon] in the battle area digivolves into this card, that Digimon will be considered to digivolve into a Digimon card with the names [Greymon] and [Agumon].

Yes, it also gains the names from (Rule).

This effect allows you to choose between playing a Tamer card with [Tai Kamiya] or [Kari Kamiya] in its name from your hand without paying the cost or hatching a Digi-Egg in your breeding area. You can either choose between these two actions and perform the processing, or you can choose to not activate the effect and not process anything.