Well first if you want to get a Girl Boxer, it would be best if your Chi is a male? Boxer females tend to get a gender thing going on if they live together with another female Boxer. But the general rule is that "Female Dogs of other breed's? Maybe it depends on the experience of the owner? So to two girl's is not a hard NO, with a different breed but it is something to bare in mind.
With a Dog and a puppy, you have to be extremely mindful in the beginning! It really does not matter that it's a Chi (kind of little-ful) it's still a "DOG." And some dogs, don't like puppys anyway and most certainly "not," puppys that are constantly in there face! Some dog's luv puppys and some dog's, not so much? It's up to you see how they get alone and to keep either one from annoying the crap out of the other. And step in if you need to? Time will tell.
And the Cat's? And Maine ****'s at that? LOL well, hopefully they won't mind the puppy? That would be good! If a cat does not like a "Dog," that Cat can make your life miserable with the Cat dog relationship for a very long time. I can/do train Dog's to get along/respect cat's just fine. But if they cat takes a dislike to the dog "out the gate?" Your gonna have to live with that for a very long time.
And some cat's luv dog's/ puppys and some cat's not so much? These days' I am down to one of each type. These days I am down to two dog's and two cats. From a high point of 14 or so cat's and three dog's. I started with cat's first and then dogs. I was into Cat Rescue and kind of sucked at it. I kept more than I placed.
But whatever I have had to rehab a couple of "potential," cat killing rescues in my time. And that never happened under my watch. I have only rule in regards to Cat's and Dog's living together. And that rule is simple ... the dog/puppy, "Never Chases the cat."
Cat's aren't stupid and understand when a Dog/Puppy has rules and when they do not. Most of my Cat's tened to luv my Dog's and they would "choose to approach them," and my dog's would either stop, dip there head's or lay down to greet them.
These day's being down to cat's Spooky, follows Bella around like a puppy, she grooms her all the time and they can in and out together. Stumpy not so much? He is an ass and often he likes to block Bella's way! He will just stand in the door way or hallway to block her at times. And Bella waits for me to make him move! Sigh he is a PIA and has no issues getting in her face when I am in bed. He totally ignores her, but if Bella rushes to the window to clear out the Helicopters ...
Stumpy always thinks she's after him? He's just an A-Hole sometimes and it's been 4 year's of the same nuttiness, but whatever.
Long way of saying, don't let the puppy chase the cat's ever. And a benefit to that is that on a walk, "your dog, won't bolt," if he sees' a random cat on a walk.
I hope you Chi is a boy you may have a chance there but there was another poster on here recently that tried mixing female boxer and female chi as a housemate and it did not work at all and they had to rehome one of them (the boxer) so I would think carefully before you do this. As far as the cats go there shouldn't be a problem and I have always had cats with my Boxers I did in fact have a maine **** I adopted from our local shelter and he was fine with the dogs.