Welcome to the area! I'd contact the Rocky River Reservation staff or the Cleveland Metroparks to double-check this hasn't changed over the years, but I remember growing up & wading through the shallower parts of the river in that area with my parents & siblings. We even have family pictures documenting one such incident of this. I'm not sure what you mean by "river stomping," but as far as I know--& again, pending verification from the park authorities, I suspect this is a place where wading, playing, & exploring could happen. However, children CANNOT be left unattended. There are spots where the current is strong, so the youngsters' play would require adult supervision AND adult physical presence with them in the river at all times.

That is correct, miles and miles that you can walk

Yes, leashed only on the hike/walk trails. No dogs allowed on bridle paths.

No entry ticket, no price for admission. This neck of the woods is accessible to the public & yours to enjoy for free.