You'll only be stamped in for the lenght of time your policy covers regardless of your visa status. If you enter visa exempt you'll only need 1 month cover.
Yes. There's no requirement to be out of the Country for a certain period of time before returning. Visa exempt can leave and return the same day or as soon as you've recieved your tourist visa you can re-enter.
Not quite. Depends on vaccination status and the Country you've travelled from. Plus you still need to fill out a passenger locator form within 48hrs pre departure.