Anytime - I love baking!
If it's the royal icing recipe, yep, just swap for water. If it's another recipe, it may want acid (acid + baking soda = lift, in stuff that doesn't have other risers), depends.
Yeah, there are generally sugar cookies drop cookies, which tend to be from thinner and crisper and more light/buttery (like a chocolate chip cookie without the chips) to the snickerdoodle, and chilled ones that you roll out and can cut into shapes which can also be thin but people often make thicker. You can chill drop dough and slice and other stuff but in general.
I'm assuming you want cookies you can roll out (based on the icing thing) - I generally use the Joy of Cooking recipe. I like their basic recipes; they work and are the classic classics imo. I've been making these since I was a kid. This is it -
Preheat oven to 350.
1 cup butter
2/3 cup sugar
1 egg
2 1/2 cups flour
1/4 tsp baking powder
1/4 tsp salt
1 1/2 tsp vanilla extract (you can do lemon or almond or whatever if you prefer)
If you want the dough itself to be a particular colour, you can add food colouring (any type) to the dough at the end. Remember the colour will fade/change a bit in the baking process.
Cream the butter and sugar until light and fluffy. Add the egg, vanilla, baking powder and salt. Mix until well combined. Slowly add flour and stir until smooth. Don't overmix. Chill dough.
It'll work better if you divide the dough into two or three or even four sections and pat into flat rounds in saran or the like before you put into the fridge. Then you have to chill for less time and you work with less at a time so it's less likely to get too warm.
Take a round of dough out once chilled and roll out on a lightly floured or sugared (you can use confectioner's sugar) surface to 1/8-1/4" thick. Cut out desired shapes, place on parchment-lined cookie sheet with spatula if needed. Put scraps into a ball and put back into the fridge while you work on the next chilled disc. You can stick the sheet into the fridge to wait if possible. If you have extra dough, or want to make a giant batch, you can wrap a disc tightly in saran and freeze for a month or so. Just thaw overnight in the fridge and proceed as usual.
Bake 9-12 minutes, until edges have a little colour but cookies are still pale. Cool on rack. Decorate when cool.
If you don't want to deal with rolling and cutting, you can form the dough into log shapes and wrap tightly in saran and chill well (if I do this, I'll sometimes stick it in the freezer for a half hour after chilling, so it's even firmer), and then slice them to your desired thickness, like you would cookie dough from the market - just better!
You can then decorate those rounds, or do stuff like roll the edges or cover the tops in coloured sanding sugar or cinnamon sugar or whatever before baking.
