Re: how to teach him to not eat the cat??
I have 6 dogs and 6 cats. It is very important to me to have a house that co-exsists peacefully together (within reason). That said, it's a lot of work. From what I can see from the above posts, paticularily the OP, is that the dog has been doing the undesired behaviour for some time, you are now trying to change that behaviour, but it is not always reinforced/followed through.
Item #1: Consistancy. This applies to the command(s) given and what you expect. The OP is using "No Kitty, Come", which is 3 commands. This makes it hard to enforce if the dog does not understand the commands seperately. "No" is generally used for making the incorrect choice. "Kitty" not so much a command but a labeling of an item, and "come", an often over used command with very many meanings to most dogs, hence it's inconsistancy.
Instead of "No kitty, come" I would use "Leave it". 'Leave it' is a universal command that simply means "what ever you are focused on when I say this, it no longer applies to you". The dog is then simply expected to abandon the object of desire. 'Leave it' can be used for all kinds of things and is generally quite easily taught. Once the dog has an understanding of the command it can be put into play with the cat. When working with a dog around a cat in the teaching phase I always have a leash trailing behind the dog. Length depends on need. A quick dog that often runs up the stairs might need a longer lead than one who jumps off the couch to the TV unit. I attach the leash to the dogs training collar. Also, have a reward handy. This can be food, a favorite toy or simply praise from you. Whatever it is should be more desirable then the object you wish to divert their attention from (the cat).
The scenario may play like this:
Dog laying in room with owner. Trailing leash within owners reach. Dog's head perks up, ears forward as cat enters room. Owner states 'leave it'. Dog either then 1) leaves it, places head back down - owner praises with reward - [but keeps dog calm]. Life resumes or 2) dog stares intently or gets up to pursue cat - owner corrects* dog. Once dog has complied with command, reward it given.
*I will not go into correction choices as there are many training methods used and that is a whole other discussion.
Consistancy of course also applies to training as well. Everyone in the house has to be on board and enforce the policy EVERYTIME. If you are not in a position to enfore the command you can not give it. Once you set something, such as a change in the dogs routine, you must stick with it 100%, or you will only confuse the dog and have poor results.
Item #2: TIMING. The OP notes that "Now we have a split staircase. He will chase the cat up the stairs, when I say No Kitty, Come he just keeps on going. He will actually come to me then run around the house and just go up the other stairs. Every now and then he will stop halfway and come to me. But when he is in pursuit he is full throttle."
It should never get to this point. In this scenario the dog is self rewarding (he gets to chase the cat) and that is far more fun then listening to you and getting his cookie at the moment. So, timing is key. Please note in my blurb above that the command was given as soon as the dog raised his head and ears went forward. This shows the dog is interested in the item (cat). This is also the easiest point in which to head off the undesired behaviour. If you are in motion at the time ie: walking from room to room, dog is following, cat shows up at end of hallway, dog bolts - you must (and this happens quite quickly) give the command 'leave it' and step on the leash. One of two things happen. The dogs listens and you reward him or the dog does not and receives a correction for ignoring the command.
You do not want this to become a game of chase (owner chases dog chasing cat) as again this is just far too much fun for the dog. You also don't want to be repeating yourself over and over 'Leave it, I said leave it, Rover LEAVE IT' and then correct, as the dog will learn that you didn't mean it the first time.
Item #3: The dog graduates from the trailing leash once the dog is 99% reliable to always respond to the command spoken at normal voice volume the first time! (1% given for a bad day). This is key. If you do this for a few hours one day and then the next day expect that the dog "knows it" and don't put the leash on and you end up with a chase came, you have set yourself up for failure and ignored item #1 - consistancy. How long this will take will depend on the dog. My boy Faeron has a wonderful prey drive. He'd happily play with anything small & furry 'to death' if I said 'okay'. When he first came to my house it took him about 2 weeks to learn that I didn't tolerate chasing of the cats. However to this day, with the cats that run if he looks at them, he is still managed. BUT he responds to me every time with the 'leave it' command. He is allowed to play SUPERVISED with one cat (a non-runner) for short periods of time, but it is ended immediately if he gets too excited. Overstimulation is what causes accidents or worse.
Item #4 & 5: NEVER leave your dog unsupervised with the cats(s) and ALWAYS give you cat and escape route. If I am not home, the cats are not allowed in the room with the dog crates. It is not fair to your dog and many cats will tease. You may come home to a busted crate and a warzone. If you use baby gates, make sure the cat can't tease the dog through the gate. I've never met a baby gate that could stop a GSD who truly wanted out. If your dog is out with you and you are working with him, make sure the cat has somewhere it can escape to should a chase ensure. Either a place up high out of reach of the dog, behind a piece of furniture or though a cat door to another room. Anything is fine as long as the cat KNOWS the escpae route is there. Otherwise it could end up climbing your curtains or knocking over all your china in an attempt to escape.
Hopefully this has been somewhat helpful. If there's anything I can further clarify on please just ask.
Dante - Mutt, Morpheus - Rottweiler, Faeron - GSD, Errol -Mutt, Teito - GSD, Pancake - GSD, Jasper - GSD, Natsu - Rottweiler, Gambit - GSD
Luna - GSD
'New Guy' - GSD