Based on this answer here it seems, that this idea that you should never let your dog into your house or room before you enter it is also bunk.
I'm just trying to confirm if that is true or not. I.e., even if "dominance theory" is bunk, are there certain behaviors**, that one should insist on for a proper relationship with a dog?
Example Behaviors:
First I'd like to clarify me opinion about the so-called "dominance theory".
The starting point of this discussion is that the "dominance theory", as it is generally depicted, is an outdated concept. For more details see my other answers on the subject,here and here.
The idea by itself, using the usual elements (dominance hierarchy, wolves, alpha dog, etc.) is a bunk. Even worse, the "theory" is used at every level of the training or interactions you have with your dog.
To me the most important think to keep in mind is that this "theory" is usually used from the start to the end: it formulates the problem ("my dog pulls on leash so he's dominant), it somehow explains the situation ("the dog pulls because the alpha dog usually walks in front of the pack) and it is somehow the not-so-clearly formulated one-fits-all solution ("don't let your dog be dominant, be the leader of the pack, etc).
That's extremely detrimental for the relationship you can have with your dog.
Dogs exhibit behaviours we dislike because of many reasons, they all do when untrained (at various levels of intensity) and there is no real point in knowing why (there are a few exceptions: medical conditions, external "extreme" stress, etc). They all do, we all need to train them.
The "pseudo-scientific" explanation is a shortcut: once you think it's a dominance problem, you don't think about other solutions and you stop looking at your dog. You start to think it's a wolf trying to overrule your authority.
The third point looks like a training method, but it certainly is not. It's a general idea we can keep in mind (of course when I want my dog to walk on leash I want to lead him, same for agility handling by the way, that doesn't need to be confrontational). Top (human) athletes projects themselves as winners. I can think the same way over and over, I'll always miss their training (and innate capabilities).
"Clicker training" or "positive reinforcement training" is not incompatible with leadership. But it acknowledge that leadership is a mindset and something you can achieve with the dog, not a training method.
That being said this doesn't mean I want to let my dog do anything he wants anytime.
As a puppy my dog was pulling on leash. I didn't really care why. I trained him and now I'm able to lead him on leash and he's not pulling. Here leading means that I decide where we go, I decide when I let him sniff ("ok go sniff the bushes"), etc.
This is my general mindset. I'll try to address your specific examples below.
I would take another approach. Nobody should want his dog to pull on leash. That's tiring, frustrating and not a rewarding experience for the human, the dog and their relationship.
Train the dog not to pull.
Then you have a real relationship, and usually smart enough humans can very easily have leadership in their relationship with their dog. That means setting the rules: we go there at that speed, we stop for a passing jogger, etc. But also understanding the dog's needs: "go sniff that bush", etc. In both cases you are leading. The rest is a training problem.
Training (that's the important word here, that means work has to be done. I doesn't mean you can dream that one day and expect the dog to comply instantaneously) a dog to sit at a door is, I think, very important.
I "teaches" the dog self control with is usually in many other situations.
It is a safety issue: a dog rushing in the house might hurt you, your wife carrying the baby, the baby, the cat, etc.
It also helps the dog understand that he has control of the environment by deferring to you! "Can I go?" the dog sits and looks at you, "ok go". That's invaluable in many other situations.
The usual "dominance" reason is "don't let him do that because dominant dogs choose higher places to sleep".
In many cases the annoying thing is not that the dog is in the couch or is not in the couch, but that the dog is trying to get in couch while you're there.
Again, pure training problem. Decide what you want, there are many pros and cons (in my case: I don't care when I'm not in the room, he can go when I'm there only if invited). First part is easy to train: do nothing, wait and see. The second is more frustrating at the beginning but in the end if you're consistent that can work (a longer answer deserves a separated question).
Yes!
But not because it has something to do with dominance.
For some trainers every "undesirable" behaviour is a manifestation of dominance. So of course yes, all these behaviours are important and all of them should be untrained (and another desirable behaviour should be trained instead).
Many training methods exists. Pets.se has already a bunch of good questions and answers about training, most of them advocating for positive reinforcement training (this list is a bit biased, I mainly looked at answers I've given):
