Well, it is because you're trying to work out how to avoid paying tax at what is probably your main residence. Are you really trying to tell us you're going to tell the tax authority where your main residence is that you're moving to another canton for a month to avoid paying the tax you owe them? Pull the other one.

I don't know, but I would guess being resident for the major part of the year, i.e. at least 183 days.

If you can afford to rent two places you shouldn't need to worry about taxation anyway, you can more than afford to pay your dues.