Karen, the issue is one of security. Luna clearly feels threatened by the puppy. To a cat, a new dog in their territory is a bit of a conundrum. Even if the dog isn't physically threatening, he could still be seen as a territorial intruder. The whole house was Luna's domain and now there's this smelly, excited being in her space. Imagine that you went to work one day only to come home to a slob who introduced herself as your new roommate. She hardly ever bathes so she smells bad, eats stinky food, and absolutely wants to spend every waking minute with you, touching you and talking to you nonstop. How would you feel?
When a cat is middening (pooping outside the box to mark territory), she's sending a very serious message about her feelings of distress. Luna's basically trying to make herself feel more secure in her territory. Cats are usually fastidious about covering their poop because they don't want predators to pick up their scent. When a cat poops out in the open on purpose, it's basically a last resort.
There are several things you can do to help Luna.
1. You need to be the police officer who decides how much contact the dog gets to have with Luna. If you see the puppy overstepping boundaries with Luna, you need to step in gently and reassure Luna while placing yourself between her and the puppy. Show her that you're on her side. Training the puppy to sit and wait are good ways to have him stand down when he's becoming too aggressive in his play. To dogs, this is just good fun, but to cats it can be overwhelming. It's especially bad when a dog doesn't pick up on the subtle signals cats send out telling them to stop.
2. You need to provide Luna with a safe zone that she can access but the puppy cannot. A tall cat tree can serve this purpose well, especially if it has enclosures where Luna can hide if she wants to. You want to think about the vertical space in your home and see if you can arrange things so that Luna can access tops of bookcases and other furniture that the puppy can't get to. This is all to provide Luna with an escape route and a vantage point if the puppy gets too physical for her comfort.
3. Never scold Luna for middening. Cats react poorly to negative reinforcement across the board. Praise her when you see her using her litter box.
4. Make sure the puppy isn't intruding on Luna when she's using her litter box. Sometimes dogs like to eat cat poop and will eagerly intrude on a cat doing her business. This would be enough to drive any cat mad. If possible, put the litter box in an area that the puppy can't access. A toddler security gate can help if Luna will jump over it and the dog can't.
5. Make special time to spend just with Luna each day to reinforce her place in the household. Make sure she gets her fair share of lap time and play time exclusively with you.
6. Be sure to clean the area where she poops with a good enzymatic cleaner. With regular household cleaners, she will still be able to smell her poop on the floor after you've cleaned it. That will reinforce her desire to continue to mark that same area.
Cats and dogs can co-habitate fairly well but sometimes they never overcome bad introductions. It isn't a relationship where you can just put them together and hope for the best. Worst case scenario would be to separate them completely and reintroduce them more slowly. Good luck!