Eeew, no "Ms. Lee", thank you! 
As for the nippy Shih Tzu, you need to put a stop to that ASAP or you risk ending up with a grouchy, nasty pup. First, let's try this: institute a "time out" system. If you don't yet have a kennel for your pooch, get one. It shouldn't be too expensive for a dog his size, and he'll use it all his life. Pick a kennel that will be large enough for him to turn around and lie down in when he's fully grown. Get a comfy kennel pad or doggie bed to put in it (the "bean bag" style is my favorite, although they usually aren't washable) and also put in some fun new toys and chewies. Leave the door open during the day and encourage him to get comfortable with it by tossing treats inside. Let him sleep there if he wants to. This will also serve the purpose of giving him a spot all his own, a "den", where he can have some alone time.
The kennel will also be for time outs. When he gets nippy, then shut him in his kennel for 5 or 10 minutes, then let him out. But the trick is, everyone has to do it; not just you. The cat can probably hold his own against a puppy less than half his weight, so I wouldn't worry much about that. (Sometimes I think my cats may be masochistic as far as being chased by the dogs goes. But who can fathom the mind of a cat? ) So give that a try, and if it doesn't seem to be helping in a week or so, I have some other ideas for you.
As for the nippy Shih Tzu, you need to put a stop to that ASAP or you risk ending up with a grouchy, nasty pup. First, let's try this: institute a "time out" system. If you don't yet have a kennel for your pooch, get one. It shouldn't be too expensive for a dog his size, and he'll use it all his life. Pick a kennel that will be large enough for him to turn around and lie down in when he's fully grown. Get a comfy kennel pad or doggie bed to put in it (the "bean bag" style is my favorite, although they usually aren't washable) and also put in some fun new toys and chewies. Leave the door open during the day and encourage him to get comfortable with it by tossing treats inside. Let him sleep there if he wants to. This will also serve the purpose of giving him a spot all his own, a "den", where he can have some alone time.
The kennel will also be for time outs. When he gets nippy, then shut him in his kennel for 5 or 10 minutes, then let him out. But the trick is, everyone has to do it; not just you. The cat can probably hold his own against a puppy less than half his weight, so I wouldn't worry much about that. (Sometimes I think my cats may be masochistic as far as being chased by the dogs goes. But who can fathom the mind of a cat?
) So give that a try, and if it doesn't seem to be helping in a week or so, I have some other ideas for you.