Here are my main issues right now:
1) She is still totally unable to be left alone in the house, even for a second. She steals anything she can get her teeth into, chews everything including the floor, and if I accidentally lose my grip on her leash, she'll run around like a maniac hunting for my cats in every corner, ignoring all commands to come back to me no matter what I offer her. I've tried keeping her tethered to me, with partial success in some rooms, but it hasn't helped her learn to stay near me at this point. She knows what "stay" means but really doesn't care.
Your puppy is too young to be left unsupervised and is also too young to have much in the way of impulse control. Keep her in a crate when you can't put eyeballs on her. In fact, put her in the crate when she acts up and let her calm down. Maybe let her have a chew toy while in the crate.
To help redirect her chewing, take a look at the types of things she chews and get chew toys that match the material. If she's chewing on the carpet, get her a frozen rag or a rope. If she's chewing on smooth surfaces, get her something smooth.
2) The biting. OMG the biting. I've tried everything under the sun to get her to stop. From screeching "ow", to biting back, to going off into the corner of the room and pretending to sob about how mean my puppy is. I know that she's in the middle of teething right now. Problem is, she doesn't usually bite during play or anything like that, or I'd just tell her to stop/disengage or crate her if she was being too hyper. Nope, she bites when I'm doing things I NEED to get done, like putting on her harness when I'm buckling her into the car, or wiping her muddy paws when we're coming inside. I know it's partly my fault, she doesn't bite hard enough to break skin but being treated like a meaty chew toy by the creature I'm dedicating so much time to is infuriating and I have a hard time staying calm with her.
Don't screech. Ever. Growl or use a deep voice for your "NO!" instead of "ow". Give her the stone eyes. Look mean and don't look away until you get some kind of appeasement (usually a quick look away from her). Don't pull your hand away, instead push it into her mouth and it'll make her disengage (kind of like "Bleh! Stop! I can't fit this thing!"). Then, the moment she disengages, change your tone, shift to calm and soft, and praise her.
Lol, by the way, I actually DID collapse into a ball of real sobbing one time when puppy Jack nipped my face and tore my work pants. He became very, very confused and concerned about that turn of events. Oddly enough, he stopped nipping at my face after that. I think he understood he was seriously hurting me.
3) Harassing my cats. I really need her to understand that they're off limits. I've tried very hard to keep her from chasing them (and they're also getting pretty good at disappearing when they hear her crate being opened). But she also barks at them, LOUDLY, whenever they're in sight. And my younger cat likes to peek around corners and stare at the puppy from afar. She seems very curious about the puppy, just doesn't want to be chased. I've been working with the two of them together on the barking. While Savannah's in her crate with me sitting next to her with (high value) treats, I let the cat come close at her own speed. Every time puppy barks, I hush her and redirect her attention to me, once she's been quiet for a short time she gets a treat. It's worked a bit but when the cat goes past the crate, she just goes ballistic, jumping and barking like crazy and ignoring me completely.
Pet gate with a little door will work perfectly. Set up a section of the house that is the cat-only area where their litter box, food, water, and scratchers/tree/whatever are. This is a place where they can escape and feel safe from the dog. Also put some kind of tree or shelves in the living room (or wherever the family gathers). This will let the cats get out of the way of the dog, and they can watch the puppy from afar. It'll take a loooong time before your puppy can ignore the cats. A very long time.
Make sure you teach the puppy "Place" or a long down. The goal will be to get the dog to stay put while the cats wander around the room. Again, I would start with getting her to understand the general goal and keep your expectations very low. You're not going to see proper self-control for a long time. But now is the time to start shaping her little clay brain so she understands the word and the action. At least she knows roughly what you want.
4) She still pulls on the leash. We've been working on this for many weeks now. It's gotten a little better but I'm considering getting her a prong collar just for my husband's sake, as much pain as he's in right now he can't handle her at all. Also it's very hard to walk her with him in any case as she focuses on me even if he's holding the leash, and if I'm walking her, I end up forced to stop a lot to correct her for whatever behavior I don't like, and if he gets too far ahead she tries to drag me to catch up with him. I hate asking him to stop every time and she gets distracted by him as well.
The below steps will have to wait until you get other commands and training shaped clearly and she's older. For example, she'll need to know sit effectively. You need to work on that, engagement, and bonding first. This means more time needs to be spent training basic commands, more time needs to be spent playing, and more time needs to be spent exercising her (not just walking, I mean running around and exploring). Throw training into play if you can by teaching her fetch and teaching her to sit/down before you throw the ball.
But I'll tell you what our trainer told us to do for an adult dog that pulls. So, keep this notation until the time is right for it and if you even need it at a future date. A martingale or a prong collar works best for this (she's far to young for either of these tools right now), but you might be able to correct with a flat collar.
1. When you are stopped, the dog sits. So, before you start walking, put the dog in a sit. Praise for the sit.
2. When you want to start walking, say "Let's go." followed by "Heel". Keep the leash loose enough to make a "J" shape.
3. When the dog gets behind you or ahead of you, pop the leash (not a yank, but a short, strong pop) and say "No. Heel." Always pair your "No" with the correction followed by the command.
4. Every single time the dog is in the correct position, you praise. If you correct and the dog comes back to heel, praise. If you don't have to correct and your dog is staying in place, you praise.
5. Periodically stop and tell the dog to "Sit" (auto-sit will come with time). If the dog doesn't sit in 2 seconds, pop the leash and say "No. Sit." Praise for the sit.
6. Repeat 1-5 as necessary.
For what it's worth, Jack did not know what the word "Heel" meant when we started the exercise. However, he's 3 and has been trained in other aspects (hence my first advice to you) so he picked up on the idea fairly quickly.
For now, train train train, shape shape shape. Teach her the actions and the words, and keep your expectations low.