Hello, I will be assisting you.
the two terms copyrighted and franchised are not mutually exclusive of one another and are generally unrelated to one another
a brand name can be trademarked. In the vast majority of cases franchise businesses also trademark their brand (this is to protect their brand as against unauthorized use by the franchisees)
Therefore in most cases a franchised business that has a brand name is often times also trademarked.
I think that's what you meant...
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My question would be is would we be ok to legally open a new ice cream shop with the name of Mr. Coolee's Ice Cream Shop without infringing on anyone?
This is a much different question than you posted. The answer depends of the name is trademarked. You should search with the US Patent and Trademark office at this link to see see if the name is trademarked) One clue whether the name is trademarked is to see if there is a circle R anywhere next to the logo/name. If the name is trademarked then you cannot use it. If however the name is not trademarked then you may be able to use the name however you would not be able to do it in the geographic area where there might be an ice cream store by that name. In some cases you may not be able to have an ice scream store by that name anywhere in state due to what is commonly known as common law trademarks
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