When she's on the leash and tries to go for the cats, she's not respecting your wishes as her master. YOU makes the rules in the house. If she wasn't behaving towards the cats on the leash, she shouldn't have been off it with them around either. If it were me, I'd have them in the same room, dog on a leash. I'd make her Sit and hold it, speaking firmly (reinforces who has control). When the cats approach or walk past and the dog starts what our trainer called "alert" watching (not just looking at them, but staring, ears up, full attention), correct her and say "No" (or "Calm" or "Easy" or whatever your chill-out command is). Correction methods vary. We've always used the rib-poke, and failing that, a short sharp tug on her choke-chain. Don't strike her and try not to be nervous ... she can pick that up and react. Once she successfully lets the cats pass without staring after them, or successfully turns her attention to you or looks out the window or whatever, immediately follow it up with praise. A small treat, clicker, or just verbal praise will help reinforce the good behavior.
It took Ebony a couple of weeks to get really comfortable with having the cats around, and even off the leash we'd keep a close eye on who was where and if it looked dangerous we'd right away say "No", make her Sit or Lay Down, and then let it be. Like I said, they're getting along pretty well these days.
I want to make sure I stress too that I'm NOT a trainer in any way, shape, or form ... this is just what's worked for us according to what our trainer told us (we had a mandatory obedience course when we signed the adoption papers). Obviously, if your dog isn't yet obeying commands or hasn't yet learned you're the Boss, that comes first. If she doesn't respect your position as the alpha, she's got no reason to listen to anything you say. If you JUST brought her home, I'd say spend some quality time bonding and establishing the pecking order
without doing "cat work". It's entirely possible she's just too stressed out by a new place and new people and new smells to handle 3 other animals who are, let's face it, prey-sized. Definitely continue crating when you're not in the house. I don't think muzzling is going to help her get used to the cats, and it certainly won't prevent her from chasing, knocking over, clawing, or pinning the cats down if she's trying to attack them.
Not sure if you've already done this, but we also found it was useful to establish a cat "safe zone". We used a child gate to train Ebony to stay out of the master bedroom, and when the cats got spooked or Ebony was getting too close to them, they knew they could hide in the bedroom or sleep there without being harassed. Eb now stays out without the baby gate, and the cats have learned she won't follow them. Conversely, Eb's crate is
her safe zone. If she's got a new toy or a treat she wants to herself or she feels crowded, she goes into her crate and the cats have learned to stay away.
I certainly hope your injured cat can make a full recovery (as far as that's possible with a missing spleen ... ouch!) and that you're able to handle and train the dog to the point she can at least coexist with your cats. Don't give up yet, just take it slow with her and remember we're all here to help
~Marrow