Honestly, and I don't mean this as harshly as it's probably going to sound, but... you need to train your dog. She's 1.5 years old but it sounds like her life has been in constant flux. You basically have a puppy who doesn't understand the rules yet, regardless of her age. Add to that she is a border collie/dalmatian cross which means she's high energy, and you have a real challenge on your hands.
She's not randomly pissing in the house - there's either a behavioral or medical reason for it. I could guess since you say she asks to be let out but half the time just wants to play, you probably stop taking her out when she asks because you assume she just wants to play... so she then pees in the apartment. It's possible she has a UTI or bladder infection and can't help it, or she could be excitement/nervous peeing, or even marking. There's a variety of reasons why it's happening. You need to figure out which one it is first and then you can address it.
The reason she does great in class is because the trainer has established a history of reinforcement for her. She performs for the trainer because, yes, she knows the trainer will feed her. She's motivated to work for the trainer because she knows that will get her something good, something she wants. Dogs do what works. Once you establish to the dog doing what you ask will get her good things, she will work for you, too.
If she doesn't do a behavior when you ask, it's likely because she doesn't
know it, despite what you might think. If she does it at class, that doesn't mean she knows to do it at home. Dogs really don't generalize well. "Sit" at the training place doesn't mean "sit" while out on a walk doesn't mean "sit" in the kitchen doesn't mean "sit" in the living room - until you teach the dog to generalize the command.
She jumps on things because she doesn't know she's not supposed to. Put her on a leash in the apartment and use it to physically limit her options. She steals things because she doesn't know she's not supposed to. Don't leave things sitting around where you don't want her to grab them until you are able to teach her which things are hers and which are not. Again, you are basically dealing with a puppy here - a baby dog. If you had a baby human, you would not expect the baby to know what things are safe to grab and which are not. You would place the unsafe things out of reach and watch the baby to ensure they wouldn't snag the stuff you can't physically put out of reach. Same thing with a baby dog. Put things up, work on teaching her impulse control
and a solid leave it,
and watch her. If you can't watch her, put her in her crate where she won't be able to get into trouble. The more she is able to get into trouble, the more she will be tempted to continue to do it. Dogs do what works - if she's allowed to steal things, it's fun, so she'll keep doing it.
Keep her away from the cat, have closed doors between them if you need to. Impulse control and a leave it will help with that too. The cat can probably take care of herself, but both breeds have prey drive and can do damage. Don't put the cat in a situation where she will have to defend herself.
The barking is going to be a thing, she's a border collie and will vocalize. But you can work on minimizing the barking
and teach her a command to be quiet as well.
For the chasing, again, a lot of this is about not giving her the opportunity to do so. Teaching her impulse control again.
You might be interested in Dr. Karen Overall's Relaxation Protocol:
I would also strongly recommend Control Unleashed
, Fired up, Frantic, and Freaked Out
, and Behavior Adjustment Training
- in that order of importance, if you only want to pick up one book. Control Unleashed holds the "Look At That" game which I think will be a helpful key in teaching your dog impulse control.
As for fetch, you have to teach that too
. Some dogs are born knowing the game means you go get the ball and bring it back - most are not. You also have to teach drop it.
I think the best way to exercise your dog, rather than trying to run her until she drops, would be to work on trick training. There are lots of books out there of fun tricks to teach, and plenty of free videos on YouTube as well. Training her to do a trick will make her use her brain, which is most likely going to wear her out faster than trying to make her physically tired. I also think you'll benefit from this because working on a fun trick will take the pressure off teaching "serious" things like sit and down (which, really, are just tricks as well) and will potentially improve your relationship with the dog because you'll be playing together rather than working and being frustrated.
I don't believe your dog is going to be a terror no matter what - I believe she's a high energy dog who has had a rough start in life because nobody knows what to do with her, so she's had a lot of half-starts, possibly some bad experiences, and nobody has followed through on teaching her the rules. I also don't think sending your dog off to a professional trainer is going to help you much, although it would give her a leg up on improved impulse control. But a lot of that is stuff that must be reinforced regularly - so you might as well learn how to teach it yourself now so you'll be equipped to work on it later.
I also think if you're this frustrated, either you haven't asked your existing trainer for help or she isn't best equipped to help you. If it's the former, definitely ask for help. If it's the latter, you might consider looking for a different trainer if possible. Somebody who might be willing to even come to the apartment and help you work with the dog there in a private lesson.
You don't have a bad dog, you just have a very large puppy who needs a lot of help. I really do recommend that Control Unleashed book at the very least. I think it will help you understand how to help her.