So the red stamps on the prints are ownership marks?
Typically the marks in red are the artist's seal. Publisher and censor marks would also appear on prints, but tend not to be red.
Why is there a cut out? My guess is it's a fan.
You are correct! This print was intended intended to be glued to a fan. Fan prints are among the rarest to survive into modern times, because they were cut from their page and glued to fans. Fans, in general, were not saved as they were part of the ephemeral fashion world of Edo period Japan.
Are those boats? Does Mt. Fuji touch a large body of water or is that just the foreground element?
Those are indeed boats. Mt. Fuji is more inland in reality. The body of water here is either Suruga Bay, to the south of the mountain or one of the 5 lakes at the base of Mt. Fuji, the largest of which is Kawaguchiko. Suruga Bay was, and remains, a renowned spot for viewing Mt. Fuji, though this print shows a bit more of an imagined view.
Why is it shaped like this.
This print was actually designed to be cut out and turned into a trendy fan. Artists, like Hiroshige, would produce new images as the seasons changed.
Fashionable people were sure to get the new designs every summer, effectively making the fans disposable. It's remarkable that any have survived to be in museum collections today!