I agree that you should have been hesitant about the treats, from puppy hood I trained Rambo (also five months as of last week) using only praise and holding his favorite toys hostage. I want him to respond because he WANTS to respond because he knows it pleases me, and because thats the "order" of things--not simply because there may or may not be a treat.
The problem with treats is that as you ween them out--especially once you stop altogether, sometimes command response will also become extinct. since you're already using treats, the best way to make sure these responses don't become extinct is to randomly award treats. Sometimes there is one, sometimes there's not...then the dog is less able to choose if she wants a treat (always having one allows her to decide the boys running around is more tempting, never having one lends to the "theres nothing in it for me" thinking, randomizing the award should almost ALWAYS work)
There is absolutely nothing wrong with using food rewards in training, and if done properly you will not to always need to have treats handy for your dog to comply. That's one of those myths that just won't die.
The rate of reinforcement should be very high at the beginning of training each behavior, then as it becomes reliable you can start to move to a variable reinforcement schedule. Think of a slot machine - we stay engaged because we never know when we'll get a reward or how big it will be. Variable reinforcement can actually make a behavior stronger because the dog tries harder and harder to get a reward, knowing at some point they're going to hit that jackpot. That can also work against you - if she sometimes gets attention for bad behavior such as jumping on you, she's going to keep trying it until it works again because there's a history of it working for her in the past. Imagine if a slot machine paid out every single time, and always the same amount - that would be pretty boring and predictable, wouldn't it? And then suddenly it stopped and you got nothing at all. Why would you continue to play?
For behaviors that are solid, (sit?), and fully generalized to all situations (will she sit in front of you toe to toe AND at your side, up close AND at a distance, indoors AND outdoors, when you're standing or in a chair or sitting on the floor, with your back turned, when she's in the car or at the vet's office?) you should be on a variable reinforcement schedule, and then you can start to eventually phase out food rewards for that behavior
. A happy "good girl!" would be sufficient at that point. I like to pair praise with whatever other reward I'm using (food or toys), and continue that as the rate of reinforcement drops.
If her sit is perfect and immediate in the house but falls apart outdoors, then the rate of reinforcement should still be very high when you're training her outside until her compliance matches that of when you're working with her in the house. Don't even think about rewarding her variably or phasing out treats entirely for that command until you're at that point.
Any time you change the picture for her by adding distance, duration, or distractions, you are making it much more challenging, and for her, it's entirely new. Dogs learn situationally and don't generalize well - YOU may know that "sit" means for her to always plant her butt on the floor/ground no matter where she is or what she was doing previously and no matter where you are in relation to her, but I guarantee you that unless you've trained it that way, in a variety of different situations, she does not yet understand that. And until she does, she's going to need a little extra help from you by upping the reinforcement under those circumstances. When I'm training a new puppy one thing I like to do is work in a different room every day - the kitchen, the living room, the dining room, the entry, hallways, the bedroom, the garage, and yes - even the bathroom! I'll sit on the office floor to train, on the couch in the living room or a chair at the breakfast table.
As Onyx'girl said, up the NILIF, be consistent, and make sure that everyone is on board with the program. You want her to understand that if she wants something she ALWAYS has to do something for you first, no exceptions.
I did read that some people put the dog up and ignore them for a while before starting this training. Do you think that this would be helpful for us?
I think you're referring to Social Isolation, but you don't actually put the dog away - it's not physical isolation, the dog just becomes invisible. I have done this once before and it worked extremely well, but it's very hard to do and I don't think you really need it. Would your kids even be capable of completely ignoring her and pretending she doesn't exist - no looking at her, no talking to her, no touching her, no interacting with her in any way? Even pushing her away if she jumps on them or telling her "no!" is attention, which is reinforcing the behavior. I did it for two days with Cassidy who we didn't get until she was a week or two younger than your girl is now. She had no manners and no training and was already a big puppy who could jump on the furniture. She did not take me seriously and thought we were playing whenever I tried to get her to do anything.
I do use timeouts when puppies get so wound up that they stop listening to me and won't calm down - it's like putting an overtired and cranky toddler down for a nap. Everybody gets a little break. Try to think of everything she does that you don't like, and how she's being reinforced for it, and then change that. Think of everything she does that you like and want her to do more of, and how you can reinforce that. Make good behavior work for her to get what she wants and bad behavior stop working. Manage her environment so she has less opportunity to practice bad behavior and get reinforced for it. If you know she goes over the top and gets out of control when everybody is outside together, your kids and the other dog, then don't take them all out together or put her on leash. The more she's allowed to do this, the more firmly entrenched the behavior will become and the harder to get rid of.
Don't expect her to come when called when she's outside unless she ALWAYS comes to you immediately in the house - put her on a long line outside so she isn't able to blow you off, and do a lot more work on your recall in less distracting places. I like to call the puppy and then run backwards a few feet, rewarding with happy praise and a treat. I usually work off leash in the house, but you could put her on leash if that works better. Make chasing you the most fun game, not playing keep away and having you chase her.
Work on self control around food and toys - sitting calmly with eye contact is what makes it happen for her. Keep treats on you at all times, catch her in the act of doing things you like, then mark and reward her for it. The more she's rewarded for any behavior, the more she's going to offer it up. This is called "capturing", and a clicker works very well with this technique. You don't give any commands, you just reward the good things she's already doing. I do a ton of capturing with a new puppy - every time she looks at me, click/treat. Every time she comes to me, click/treat. Every time she lays down, click/treat. Build the behavior you WANT, don't just try to get rid of the behavior you DON'T want. Acknowledge her when she's calmly laying on the floor chewing a bone or playing with a toy: "Good girl, chew your bone!" BE FUN!!!