Since the first pets came back, do new pets thrown in the rift still take a few days to return? Or do they come back the next day?

Unsure; I know some folks waited about 48 hours, but mine was back in less than 8. I wouldn't be surprised if it was variable on purpose.

So you don't even have to visit it again for them to return? Do you get a notice?

Nope, it just reappears, with a sidebar event to tell you that it's back.

Yeah, the pet will just return on its own whenever it's ready in whatever amount of time, and you should get a sidebar event letting you know it's back.

My pet also stayed less than ~12 hours. I don't know the exact time he came back, but I threw him in around reset on whichever day and he was back before I got online the next day

Great, thanks :)