Anyway, in the process, a big frog shows up and starts swimming around
(coming out of hibernation?). Well, I brought the frog inside and now
we're expecting more really cold weather.

Since you've already brought the frog in, and you're not sure what type it
is, I'd keep it in till a good warm day arrives so it has time to get where
it needs to be. Not all frogs hibernate at the bottom of ponds. ~ jan