Litter tray, clumping cat litter. Put a piece of used paper under the litter and another over the litter, so he smells that this is a place he's been before. Praise and once he gets the hang of it, get rid of the paper and just stay with the litter. You may have to clean it often, I have one who won't go if there's more than two pee pee clumps, another that doesn't care if I ever clean the pan.
Until a puppy is at least 6 months old, they're not reliable. Also, having him neutered ASAP will help him learn not to lift the leg and mark all over the house. It will also make him calmer, which will help the next issue.
Well, that has got to stop. I suspect he's just a little jumpy, dealing with the loss of his old owner, getting used to his new owner and a new place to live, etc. Try the introductions again, but this time you hold Buddy so he is secure. Also, have your friend reach palm up and pet his chest rather than come at his head from above palm down. Pet and praise him for being good, growling or etc. should be immediately dealt with by a stern "NO". Never hit your chi, he won't understand, better to chuck him gently under the chin or 'scruff' him by lightly grabbing the fur at the back of his neck. You can't pick up a dog that way, as you can a cat, but that's how a momma dog tells her pups that they've just done something wrong.
It may take a while for Buddy to become friendly, chihuahuas are very much a one or two person dog, it takes work to socialize them to accept lots of people. The best way to reduce snapping and etc. is to introduce Buddy to lots of people, again going slowly and with him safe in your arms. Buddy may also express jealousy of other people and/or other animals, perhaps the cat startled him or he interpreted the cat as trying to take his place with you. Pay attention to both animals, you have two hands and two pets, you can figure it out.
The odds are that Buddy will challenge them, chi's have no idea how small they really are. I suggest you introduce him to the dogs, make sure he's safely in your arms, and never when there isn't someone right there watching. They may get along, there may be some growling and snapping but don't let that concern you too much, having Buddy neutered first will help quite a bit. As long as there's no blood or loud yelping, they're going to work out the hierarchy by themselves. If Buddy just doesn't get on, then he'll have to be kept separate. I wouldn't worry about puppies once Buddy is a year old, most chi's will accept a lot from a puppy, they're pretty tolerant.
One last thing, I recommend the book Chihuahuas for Dummies. It's the best book I have on the care and training of these adorable little creatures.