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In the Villages card game, we did not understand a few rules.
Answering the questions in order:
Can you pick up cards from your villages during your turn?
No. when a card is played to a village, it remains with that village until either it is destroyed, it is kidnapped, or all units of that village are destroyed. The singular exception (that I am aware of) is a guest unit in an Inn - by the Inn's effect, that guest unit may return to your hand at the start of your turn.
[edit: The "Distant Lands" expansion adds a second building card that allows two additional guest units (one of each color adjacent to the village's color) to be in the village, and like the Inn, they may return to your hand at the start of your turn.]
Can multicolor/light units switch which village they are in if possible?
As noted in the previous question, the only unit that could "switch villages" is a guest unit residing in an Inn.
if there is a village made of light units, what color is that village?
A village's color does not change, even if the only remaining units are multicolor or light units. The rules do not specify whether or not a red village with only Light units remaining is considered destroyed or not, only that there is no such thing as a "Light village" (that is, when the village is created, there must be at least one non-light unit in it)
According to the description of units in the official Villages Wiki colorless/light units can fill in for any color. This includes being part of the Village founders. For example:
Or
However they (the light units) cannot form a village of their own.
