When I bid on a FA in mid-week stage and am successful, do I have until the next day to cut a player to get down to 53? Or do I need to cut a player in the mid-week stage as well to be sure I am at no more than 53 if I am successful?

Thanks

The roster limit and salary cap are not enforced on the midweek stages. That is actually the purpose of the midweek stage, to sign players with enough time to get your salary cap back in order.