Stating the obvious, not so nice thing that you don't want to hear
Reading through your post, I think the biggest issue here isn't that you're having issues with the dog, but that the dog seems an ill fit for your girlfriend, who is its owner. I really think that this high maintenance dog requires a skillset that's beyond either of you combined. And I think the best way to solve the issue is to give the dog away and hope its next owner does have those skills - and the dog will be happier where it doesn't have to be so viligant. There is a lot to be said for matching personalities/skillsets and this dog seems badly fitted to your girlfriend and evenmore so to you. I'm really focusing on your girlfriend here because she is the dog's owner and master, and if she can't handle him, there's little hope for you. I think that your best option is to do your research on a dog that will suit both of you- something that's not so guarding, barky, ignores cats, and preferentially laid back and low maintenance, and I think the quality of both your lives will be significantly better. Or maybe no dog at all. This is a mean thing to say - but I have found that when people own dogs like this, they really aren't attracted to the dog much at all, but rather, the dogs appearance, and the few cute mannerisms it has - everything else that the dog is, the majority of who the dog is, they are not happy with at all. (people tend to pick dogs that look like them).
Consistency and girlfriend
Dogs are supposed to be a team effort and consistency is key. You haven't written much about your girlfriend and that lack of it leads me to believe that she really isn't so much on board at all. It's always you trying this and you trying that as if you're the only one that has an issue - pretty sure the dog acts up when you're not home. If this really is the case, then I'm afraid this is about the best its ever going to get. Either the girlfriend gets entirely on board, or this is your life, now. Half-assed will not do. Your relationship with your girlfriend is out of scope for his question, so I'll stop talking here.
Master of the house
I also wanted to mention that it sounds an awful lot like the dog is the master of this family, and not either of you humans.
On addressing problems before they become problems
I don't know what training methods you are using, but if it were me, I'd be working on preventing problems from ever happening. It sounds as if you are just addressing symptoms of issues (maybe not, dunno what your training is like) and not the root issues themselves. With jumping on the table, the dog really should never be jumping at all, and it should never be taking food that isn't his. With getting mad at the cats, it should never be paying attention to the cats, period.
You have to understand that your sanity is far, far more important than how comfortable the dog is. I feel like all the measures you take are moderate. Please never be afraid to go to a more extreme solution if it gets rid of an issue. For example, what if you crate the dog at night? Then he cannot get out, and you get to sleep in the same bedroom. It's not a moderate solution, but it gets the job done. Because sleeping in two bedrooms because of the dog sounds as if the dog is completely terrorizing your relationship. Things should never get that bad. Your life sounds like a reality tv episode where a dog trainer comes in and fixes everything - even with the improvements that you've made.
Dog training classes
I know that you have a personal trainer already, but I would really suggest both of you going to dog training classes with your dog, even if he already knows basic commands. I feel like it will help you to bond better with your dog, help you get the right kind of attitude that the dog can respect, talk with other owners, and help you figure out how to manage and control your dog better. My dog already knew basic commands and had no issues but dog training classes were the best thing I ever did for her - and for me, too. I really recommend looking up your city's local kennel club for class rates deeply discounted from the rates a pet store would charge.
On attitude, attitude is huge. It's hard to describe what it is, but it's basically a changing of yourself (in a good way). I realized I was jittery, anxious, and jumpy, and all this my dog reflected and magnified into a very scared, unhappy animal. I learned how to be calm, collected, take charge, etc. Next time you are around dogs, take a look at how an owner acts and how the dog acts. Make some observations, and then think about yourself. Attitude is going to be one of the biggest improvements you can make.
Schedules and rituals
Dogs almost need schedules to be happiest and spur-of-the-moment / fly-by-the-seat-of-pants families have the hardest times, especially with high maintenance or energetic dogs. Adopting a schedule and keeping it that way can greatly help. I'm not suggesting you change your life all around and make everything up anew, but rather to try to keep doing what you're already - but just keeping it the same way.
Many times dogs don't have a clue what they're supposed to be doing and do whatever they think is best. So if the dog knows what he is supposed to be doing, there isn't nearly as much issue. Establishing proper protocols comes from those rituals.
For example, doorbell rings? BARKBARKBARAKRAKARKAKARKAARK... it's a crazy house in here! Establish the ritual by telling your dog what he is supposed to do. (Go sit or lay somewhere and be silent). Most times dog owners only tell the dog what he is not supposed to do, and it doesn't work well because it is a process of trial and error for the dog. It's too complicated, and too stressful. (You can always practice rituals, too. Say, you have girlfriend ring the doorbell and you practice with the dog inside).
Dinner time? Maybe dinner is around 5 and the dog should be laying on his bed. Or maybe during dinner preparations / cooking the dog should be laying on his bed so he isn't underfoot doesn't get stepped on and isn't begging at the table .
Couch time? Perhaps the dog is supposed to be only in one spot, so that he is not around the cats favorite spots and interactions with the cats are limited.
You pick whatever works best, enforce it, be consistent, and often your dog will be happier because now he knows what he is supposed to do, instead of doing everything all the time randomly and intermittently getting fussed at for things he doesn't understand. The theme here is that you can create rituals that circumvent problematic behaviors before they start.
Rewarding behavior and not rewarding everything
On this note, be very particular with rewarding. You never want to reward your dog for doing "everything". Often, I will tell my dog to sit, and she does all the tricks she knows, including sitting. This is not the desired behavior, so I don't reward. No treat, I walk away, and "punish" her by ignoring her completely (surprisingly effective); I don't reward her with more attention. I try again later, and only when she sits and only sits does she get the treat. There is no "feeling sorry" for her and giving her the treat anyways, this is very detrimental.
Rituals and moods
You can also have a mood be a part of a ritual! In fact, it is probably a ritual already.
Example, you open up the door to come home from work, and there is the dog. Many, many owners greet their dogs with WHOSAGOODBOYYESHEISYESHEIS , and can't understand why the second they open the door the dog goes into a hyperactive looney madness! They have trained the dog to be in high energy mode when the door opens.
Walks are another great example where dogs are often hyperactive. A walk is supposed to be a relaxing experience, but it's often instead your dog trying its hardest to give you a heart-attack and simultaneously de-socket your arm joint. It shouldn't be that way! Start out with the mood you want and re-inforce it. Dog bouncing around everywhere? "No, sit". Only when the dog is calm does the leash go on. Dog pulling and going nuts? "No." Only when the dog is calm and not pulling does the walk continue. You may have to stand there for a while an idiot, but it works.
Same with around the house. You can establish a command like "Settle down", and reward only when settled. It's entirely possible to create a dog that is mellowed down three notches. Settle him down when he should not be riled up and watch your dog become happier, as if he is saying "Whew, all that bouncing around sure was stressful to me, it's so much better now I know how I'm supposed to act". Petting time doesn't need to be a vigorous commendation for saving little Timmy from the well, it can be a calm, relaxing, methodical rub. In fact, your dog may be happier like this. Homework assignment for you both: See if you can get your dog to nearly fall asleep while being petted.
Remember that your stress contributes, so think about what's happening. And get the girlfriend on board, or else it's basically all for nothing.
There's a ton of content and you're not going to be able to swallow it all at once. Here's a list of homeworks for you:
1 - I would really like you and your girlfriend to watch some Caesar Millan's The Dog Whisperer; there are some episodes on youtube. Some people hate him, but I'm recommending him because I want you to see how the behaviors/environment in the house have created the dog. I want you to see even more how radically things can change and get better.
2 - Talk to your girlfriend about consistency and getting on board more. Seeing those videos should help you both feel more empowered and inspired that you do not have to live this way
3 - Start thinking about your problems and how you can fix them, talk about what will work for both of you
4 - Have a session where you lull the dog into near-sleeping by petting alone. Have a session where she does the same. I want you to see that your dog does not need to be a tazmanian devil (unless you want him that way).
5 - Investigate dog training classes, which will also be inspiring and empowering, but more hands-on. It is hard to describe the can-do aura and attitude that is lit there.