Yes it's legal at all times just needs to be a section line or other marked trail. Some areas that are owned by the government are not usable trails anymore so keep an eye on that.
Dogs running around chasing deer are no different than a coyote. I would have a hard time pulling the trigger on a dog unless it was aggressive towards people or other pets.
I was once told by a wise hunter that a ditch tiger with a collar is just like a goose with a band.
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And yes I have 2 cats in my house.