Outstanding start in the right direction Roxanne.
Let me give you some feedback, from my perspective, on your blow-by-blow account:
1. sign language: you discovered all by yourself something many dog owners never seem to learn, that dogs have a much easier time learning signs and body language than they do have in learning 'words'. Good going! That is why a good trainer will always first teach the sign for a command, and then introduce the verbal command. A dog should, in the end, always know both ways and react to both when given by themselves.
2. Very good! Yes, right now (once she has all the rules in her little head, one can ease up a bit - its like with kids really :>

not too much in life is free for her.
3. Don't expect this to work always as fast. Make sure NOT to use her crate as a place for punishment (that is, time outs). Normally you would use a room, or a hook at some kitchen cabinet or a leash at a door knob and so forth.
4. That is a good idea. However, try not to be too harsh in withholding atm - that can discourage a young dog very fast. Normally a 30 second delay is plenty in my experience.
5. Well, it cannot hurt to even handfeed the dog for a few weeks if there are serious problems. That often helps quite well in establishing who is in charge.
6. Outstanding! Eventually, you can take this a step further. Your goal should be that you order your dog to rest where you want her to rest while you guys eat. If she does that nicely, she often gets a treat afterwards. So, if tomorrow she again lays down nicely after she notices nothing will be handed to her, try to capture this behavior and say a cue (for instance "Princess, settle") right when she lays down. After dinner, she then gets a dog cookie. After doing this routine for quite some time, you will be able to say "Princess (or whatever her name is) settle" before dinner, and she will know what to do in response.
7. Very nice.

I think it is important to understand one thing here however: it is nice to go through a door first, because it often times is safer for the dog. As a matter of leadership as which this is often touted, it is complete nonsense. You see, what is important is something the leader decides and is not cut in stone. If the leader cares to be first through a door, then he will go first. If he cares to go last, he goes last. If the leader wants to drink first, he drinks first, if he does not really care, he drinks whenever. And so on.That is, I believe, an important lesson to always keep in mind.
8. Perfect!
9. Giving the dog less attention to become "alpha" is, in my humble opinion and experience, complete hokus-pokus. Given that, so far, I never had a problem establishing the leader position while woefully disregarding that piece of advise, would lend itself to put a chink into this sort of reasoning. Remember what I said in point 7: the leader gets to decide what is done and when it is done. If you decide to call the dog over to pet or massage it, then that is what will happen. So please, give your dog all the love you want to give, just make sure you give it when you want to do so, not when the dog solicits it. If the dog solicits it and you want to give it, wait until the dog gives up. Then, a minute or two later, you initiate it.
Rewarding her randomly for good behavior is an outstanding idea. You should play as much as you can with her, as play is an excellent occassion to establish many rules. For instance, if during play she accidently nips your hand, you yell 'ouch!', get up and move away. Play stops for a minute or two, then you can resume. If you wanted to stop playing anyhow, you just stop it right there with your 'ouch!'. If, so far, she was used to nipping, you should not be too tough on that. Make a difference between hard and soft nips, first marking and getting rid of the hard ones, then later of the soft ones.
If you like your dog sleeping in your lap, call her over and put her in your lap. I think you get the point by now.

(and btw it is true: a dog living with clear rules and a benevolent leader will be the happiest of dogs)
Your ideas about coming when called, however, clearly show the misappropriations the "alpha theory" is so infamous for and that you probably read somewhere on the internet. A dog coming or not has absolutely nothing to do with your status - nada, zilch, niente. A dog looks out for number one - herself. [

] So a dog will come or not due to a simple calculation: what is better for me... to come or not to come? If every time the dog came to you there would be a good chance of some fleeing squirrels and rabbits right by your side, with a huge steak after the hunt, the dog would come like a rocket every single time, even if you are in her view the runt of the pack - trust me. [

] Alternatively, if you use a electric collar, the dog would also come every time - simply to avoid the punishment - and again it would not matter one bit if you are a leader or just a rock at the beach which has proven herself to be a safe place. If a dog is trained properly with positive reinforcement methods, the calculation will almost always tell her to come. However this is a long training process, described in detail for instance in "The Culture Clash".
Commands giving once, as already said, is a good idea. With a young dog you can do a little alteration of this however. If you say the command once, and she does it, she gets praise and often a treat. If she does not sit, for instance, on the first try, you go a step closer and repeat the command a second time, maybe in a bit a sterner tone. If then she does sit, you say 'good girl' but there will be no treat. If still she does not sit, you go over and gently help her to sit. Once she sits, say good girl and release her. A little later you try again. You see, you do NOT want to teach her that she can ignore your commands, which you might be in danger of when simply ignoring her 'ignoring you'. If you say sit, you mean sit.
My congratulations on all your swift progress. I am absolutely certain you, your family, and your dog will all have a great future with one another.