What you are seeing is all perfectly normal. The older dog may be a little rougher than if she was just a little older, but what you are seeing is exactly what older dogs are supposed to do.
Dogs teach puppies manners and the way they do that is to growl when a pup is doing something it shouldn't do like approach their food, etc, and when a pup fails to listen to the warning, they lunge sounding ferocious and usually will even snap their teeth together. They might even nip but usually do not do any real harm to the puppy. It is more of a scare tactic toward the puppy. All dogs will usually do this with puppies.
Often they will allow play behavior and rough housing but if a pup should bite too hard, they will turn and reprimand the pup with the growl, lunge or ferocious behavior. This is in the pup's best interest. A dog in the same household is going to be much gentler with them than a strange dog would be. In additon, pups get away with anything when with their mom and then around 4=6 weeks she starts teaching them not to bite hard and what some acceptable behavior is. However, they leave mom and still believe they can approach all dogs like they did their mom and can bite ears, scratch stomachs and even try and nurse off of them in some cases. So adult dogs reprimand unacceptable behavior but they do take age into account. Most will give a puppy lots of leeway in behavior but each successive day, that lessens and lessens as the pup has to learn acceptable manners around adult dogs.
So as long as blood isn't being drawn, it should be fine. Now if the pup is also female, you might with to get her spayed before her first heat cycle. The current adult might feel her place in the household as the top dog might be threatened and then you will see some aggression and more serious reprimands that might indeed draw blood. Spaying will help reduce the chance of that occurring.
You shouldn't have much of a problem with a male since there can be both a female and male boss and he'll likely see her as the main dog since he was raised around her.
You can help teach the pup as well by going ahead and starting obedience training. Obedience training will allow you to call him to you if you see him starting to do things to annoy the older dog. It will also keep him occupied for longer periods of time and give her a break from constantly reprimanding him for unacceptable behavior.
Structured classes are great, but you can do structured training at home long before a dog is able to be out in public around other dogs. The following site is helpful in helping owners train their dog. Be sure and click on the link to the page on obedience at the bottom. and links on subsequent pages leading to detailed instructions.
Training works best if you train at least 30 minutes a day (two 15 minute sessions). I would start making your dog work via the Nothing in life is free program (NILF). It is outlined below.
It also sounds like Kate could use some refresher obedience training as well. You should feed Kate first and then the pup. Once both have some training done and are listening pretty well, you can feed them in the same room. However, start by isolating the pup, then fixing the food, and put Kate's down first before taking the puppy's food to him/her. Once you try together, keep a leash on the aggressor and slip it under your foot so Kate can't attack the pup. GIve her just enough room to eat her food and be sure the pup's bowl is across the room. I've had some owners put an eye bolt in the floor on one side near a wall so they can clip a dog's leash to it to hold them in one area of the house. However, even using these types of restraints, you need to be standing in the middle and if either dog moves toward the other for any reason, give a really low toned "NO" so both dog's know it is unacceptable. Eventually you have to remove the restraint but if you have keeping up on the obedience work, you should be able to have them lay down or sit in front of their bowls and stay even if they want to move toward the other one.
Obedience training can also show the older dog more attention and stop some of the jealousy issues we see from an owner spending so much time with a new puppy. We all do it since pups need so much care, older dogs can feel left out and will take that out on the puppy. So start obedience training with them and let Kate continue to help the pup learn manners, but if it gets out of hand, let me know and we will work on some other techniques. I do believe that obedience training will help a lot and prevent worse problems from developing as the pup matures.
I hope this information is helpful to you. If you would like any additional information or have more questions please don’t hesitate to press the reply to expert or continue conversation button so I can address any issues you still have . If you do find this helpful, please take this opportunity to rate my answer positively so I am compensated for my time.