I roll my cookies to a 1/4″ using wood dowels on the sides of my rolling pin to roll over so all the cookies are the same thickness. I have a whole post on cut out cookies here https://www.createdby-diane.com/2017/09/how-to-bake-perfect-cut-out-cookies.html that answer lots of questions I’ve gotten over the years on cut out cookies.
I don’t often bake with other flours, but gluten-free flour should be fine as a replacement in this recipe.

2. Do you think you could switch out All purpose gluten free flour by any chance?

The best tip I can give you with the info you gave in the comment is to keep mixing the dough, it will pull away from the side of the mixing bowl and come together nicely. If you tend to roll out cookies with a lot of flour, be sure to reduce the flour in the recipe. I now prefer to roll out my cookie dough in between wax paper as I don’t need to add any flour. Hope that helps!

if the dough was soft, I would say it had to do with the temperature of you butter. These cookies do not have a really soft consistency, the butter should be left at room temperature for about 20-30 minutes. If the butter is too soft and you are able to press you finger into it, as with most cookies they won’t turn out well. The other issues could be if you oven runs hot, you can easily test that with an oven thermometer. My guess would be those two things. I use this recipe often with no issues of browning or softness, so try cooler butter and the oven should be at 400, if they are done sooner I would remove them before 7 minutes, not lower the temperature, test by baking two cookies at a time until you find how you oven works best is the easiest way to test cookies, not bake a whole tray if you are testing oven temp and such. Hope that helps.

Yes, I use an electric mixer to cream the butter and sugar together. My arm would be exhausted trying to mix it by hand.