No need to stir, at the very most you draw liquid from the bottom via a valve at the bottom/side of the barrel and then pour in the top; this supplies oxygen and is ideal to do once a week, but most people do not bother.
The mother??? Raw Vinegar is "ALIVE", the mother that forms on the top is not needed, the more you throw it away, the more it will grow back. Some may suggest to eat this mother and the books suggest you can and you might be OK and again you might not be OK, so I suggest it be pitched.
Raw juice makes its own mother, you don't need mother to start a vinegar. After a year the mother gets so thick you can cut it with a knife.