At least here in the U.S., if you listed, as stated, as both Auction with a 'buy it now option', and someone clicked on "Buy it Now" in your listing that would end the sale and would remove it from listings, "auction" or not. Then, like any other sale, you would await payment before shipping...
There's a difference, of course, between "Buy it Now" and constructing a listing that included "Best Offer". In that case, someone could make an offer (below what you have the opening "auction" price as), but you would then have the option to "Accept", "Decline" or make a counter offer. This is different from what I think you're describing though...