I'll apologise for length now!
I've worked with many many dogs, i worked in rescue with everything from adorable puppies to killing machines on death row. I have rescued many dogs privately that would have not benefited at all from being in rescue centres as they needed one on one attention to address behaviour issues, and sadly in a some cases mental problems. All of these dogs left me to their new owners as good kind dogs. I did home checks and got veterinary references for all of them, and did not make any money from it at all. I have my own 2 dogs that never put a foot wrong and the rescue dogs did well with us as my dogs were "leading by example". I have an 8 year old rough collie cross who has been with me all her life, and a 2 year old ex breeding rough collie who had never lived in a house before.
I'm not stating this to imply I have more experience or know better than anybody else here, I just thought I'd give you some background on what experience I have and how my methods have worked. I'm always open to new ideas and love hearing other peoples tricks/tips and advice!
I think the pack theory does have a place and that respect is something you should earn from the dog by the care you give and being a calm, confident and consistent leader. The dog should do as you ask because he respects you and trusts the choices you make for him. These dont have to be reached through negative reinforcement, infact something like agility classes work very well.
I also believe that prevention is better than cure in a lot of cases, for example if the dog is stealing from children he gets his "reward" because he gets the food, ignoring this would only lead to him doing it more as he got older/bigger and obviously it will make him pushy around the children, dogs aren't stupid and he will soon learn what time is meal time and perfect the art waiting until your back is turned. At this point, telling the dog off would be uneffective, as he has his reward from the food and wont understand why you are annoyed. It might be a good idea to teach the a dog to go to his bed on command instead, and have the children eat at the table, when no food is forthcoming he will eventually learn to ignore it entirely, if he is allowed at a young age to take from children he will continue to do so. Quite simply a dog that has never learnt this behaviour wont have any compulsion to do it!
I also never use treats when I am training as I think it makes the dog only work when he is being bribed, rather than doing it because he is asked. Sometimes dogs can even learn to repeat negative behaviour (such as running off) as they get a reward upon return. Saying this, I also have dry food down constantly and my dogs eat as and when they need it, I know this wouldnt work for some dogs but it suits mine very well and combined with other methods can be one of the best ways to stop food aggression. There is no need for the dog to worry about losing it, having it taken away, protecting it, as its there all the time. I found this to be very effective in helping a very thin collie i rescued who when i got her would have bitten over food. She calmed down considerably when she realised that food was something that was a given and wasnt about to be stolen from her or about to run out.
The idea of a "pack mentality" is something that i'm afraid is in all dogs, they learn this from a young age when pushing for food with siblings and they quickly develop a pecking order amongst themselves in multi dog households. It is not meant to be an oppressive way of understanding and controling your dog as it has been used in the past. Most dogs learn where they sit in a group and as long as everybody is happy then there should be no negative behaviour from it, a pack is about a co-operation, its how these animals survived for many years and is in all dogs. It can be witnessed in the most simple of gestures, such as my older dog making sure she is the first to the water when we come in from a walk, the others wait turn as she is above them, she doesnt sit higher in the pack because she is given special treatment or because she is aggressive, but because she shows herself to be calm and fair with younger dogs we have rescued.
A problem for some dogs is not being allowed the chance at being taught limits, this is often a problem for dogs that have been seperated from their mother and siblings too early. The mother will use sounds to tell her pups when enough is enough. Puppies that arnt exposed to this often struggle more in training as a result and my have mouthing issues. It is also important for the pup to spend time with siblings as they learn the limits of their mouthing by nipping and thus upsetting their siblings and then being told off for it. This is hugely important to a dogs social developement. It is expressed in play at this age and is essential to a dogs mental growth and understanding of boundaries.
The best way to make sure your dog understands the boundaries in your home is to set rules and never allow them to be broken. Dogs love consistency and if you put a rule in place (such as the dog not being allowed on the sofa) you must make sure that nobody else allows this. If it has never been allowed it is unlikely that the dog will even consider trying it after just a few weeks. If you do want to use a word to express that you aren't happy with the dog (this is entirely up to you) I would recommend a LOW tone, a simple "no" or "Oh dear" in a low voice will get your point accross far better than shouting or screaming that he wouldnt understand and may leave him scared or confused. Low tones work better for this kind of thing as mother dogs use a low grumble to tell pups when a behaviour is unacceptable. If the pup is jumping up use the low toned "no" and dont make eyes contact, keep your hands to yourself and he will understand you perfectly.
It's hard to give you any advice regarding things that would help to make sure he respects your DCs as I'm not sure of ages and obviously how a dog is expected to behave around a toddler is completely different to a teenager that would be fine to walk him and spend time training him. I would advise though that you dont allow the children to give the dog a treat from their hand, instead place it on the floor and then allow the dog to take it. This will pay off later as the dog wont associate your children's hands with food!
Apologies again for length.... I got carried away!
Just remember training a dog takes a lot of time and understanding, and you sound like you are doing really well so far!
Different approaches work for different dogs, I hope this might be of some help.