I tend to add a little more water with powdered sugar. It sounds like your powdered sugar is not fully stirred to dissolve before shaking. Sugar syrup is much easier and more consistent!
We do list gluten as one of the 14 allergens, but as Michael says, gluten is not in any of the ingredients above. Some bottles of vodka etc., say they are gluten-free, but all vodkas and distilled spirits are gluten-free.