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As for the flour, changing it to gluten-free might not form the way you like. But you can always try and come back with the results. Or google a gluten-free dessert menu :)
Another issue with carrot is that it doesn't hold at room temperature as well so unless you plan to eat/freeze the cookies within around a day the carrot within the cookie will begin to look somewhat unappealing.
As for flour, most gluten free flour can be used as a substitute for most recipes 1 to 1 if you add a binding agent alongside it. A 1/4th teaspoon of xanthan gum or tapioca for every cup of gluten-free flour will usually suffice as an extra binding agent.
I'd personally not use carrot for cookies but if you think it would be great. I would say go for it.