I am new at this, but how melty should the sugar be? I'm also concerned about the egg whites cooking.

Sugar doesn't "melt" until it gets up above 350F (actually closer to 368F), which is more than high enough to kill off anything worrisome. And should it somehow miraculously survive, your cookies are then going to spend some 45 minutes in a 275F oven. An egg, like other foods, need only hit 165F to kill off bacteria. So you need not worry about the egg whites being cooked when you make these cookies, nor with food poisoning somebody. And so you know for the future, sugar is actually a pretty good antimicrobial agent. That's why it's used as a preservative, and keeps your jams and jellies safe. Oddly enough, sugar, usually in the form of honey, was used effectively since ancient times to treat and help wounds heal. So long as you follow the recipe, and maintain proper food hygiene practices, you will do just fine and the cookies will be quite safe.