They can have swimming water after the first day. However it has to be shallow enough they can get out easily. If the get waterlogged and can't get out they'll chill and die. They'll swim a little then hurry back to the heat to dry themselves and warm up.
Yes, but if the swimming water is in the brooder you are going to have to do a full clean of the brooder much more often, they need to be able to be warm and dry and that swimming water will end up everywhere but in its container.
I think it would be better to have the swimming time limited and outside of the brooder, or maybe right before you plan on cleaning it anyway.
They can start swimming by day one, but again, limited and given time to be warm and dry
They can have swimming lessons from day one on. In Nature one of the first things mama duck does is guiding her ducklings to the nearest pond/lake/creek/river for some swim time. In nature the ducklings get preening oil from mama duck and are water-tight to some degree. You must monitor them during swim-time and get them out when they are getting too wet.
I wouldn't put any other water into the brooder other than drinking water. and limit the access to that so that they can only get their heads in. Otherwise you will have to clean out the brooder several times per day. (Unless you have a 2,000sqft brooder…)