I think you're spot on in your analysis of Akira (awesome name, btw!) and her reactions to your husband and kids. Barking is good in that situation. I mean, do you want a dog who warns you that's she's scared, or a dog who goes straight for the bite? I know what I want!
Here's the thing: this isn't about Akira. You know what to do with her. The problem is your husband. He obviously wants a certain dog and Akira isn't that dog. My new dog, Kabota, wasn't really what my husband wanted at first, but he learned to appreciate Kabota's finer points, and to work around Kabota's problems (he's a very soft dog, like Akira. You do have to treat a soft dog a certain way to bring out the best in them.)
If your husband is willing to be patient, to give Akira time to adjust, to treat her the way she needs to be treated, I'm sure she'll be a great dog. If he's not, I'd rehome her somewhere where she'll be treated as she needs to be treated.
I agree with Amaryllis. Also, keep in mind this is a baby, at 13 weeks, and she is very new to your home. She is still adjusting to being in a totally new place with new smells, sights, and sounds.
Anytime you can't physically be right there to supervise the kids and the dog together, the dog should be somewhere else. Crating is absolutely ok for this situation, when you need to get things done, and can't be right there. You can give the dog a filled kong (freezing them overnight makes it take longer for a pup to get the good stuff out - we use peanut butter). This might keep the pup busy so she doesn't whine or cry.
Also, teach your kids proper dog manners, the same as you are trying to do with your husband. You're right, they're just kids, but they can be taught (you probably already are!).
In your situation, I think it's really about managment, managing your dog's environment, to set her up to succeed.
And, check out some of the stickies, they have great information!
I really don't know what to do about this peeing thing...
Yesterday my husband came down the stairs, and as she saw him coming, she ducked into a different doorway to get out of line of sight of him/out of his way. When he got down, he stopped, she came up behind him, sniffed his legs, and circled him, as she circled him she peed all the way around... my husband didn't even touch, speak or outreach an arm to her. Later that night before putting her in the cage for the evening, we had nearly the same experience, he came in, stood still, she came to him, he spoke to her gently and layed a hand out for her to sniff (she came to his hand) and she squatted and peed while looking up at him submissively.
This morning, I let her out of her cage, again.. not going "HI HI HI OMG HELLO HI SWEETIE!" to excite her, I didn't say anything, but simply opened the door. She was whimpering slightly as I did so, when she came out, she started to jump up, so I turned away from her until she sat and looked up at me. Upon doing this I bent down to praise her and give her the affection she desired. She was so excited, she peed.
I've never had a dog that did this... I'm really hoping this is something she will grow out of, any thoughts?
Also, how best do you tell a dog what they've done is wrong? Last night she was out in the backyard with myself and my two kids. The kids weren't playing with her and were doing their own thing, I was playing with Akira trying to get her exert some energy before bed, I got her excited and while she had not been paying any mind to the children, she ran towards my 4 year old and jumped up onto him (he was just standing in the yard pulling petals off a flower) she scratched his back and belly up and made him cry. I yelled at her sternly as she ran towards him in an attempt to discourage her from doing anything stupid (it didn't work) when she lunged onto him, I grabbed her by her collar and spoke to her sternly, pointing at my crying child and saying no. I then lead hey by her collar into her cage inside where she stayed for about 30mins while we got the kids ready for bed.
I cannot have her doing this to my children. They said they don't like her and she is a bad dog as a result of last night. I tried to explain to them both that she is just a baby and that she has to learn not to jump up onto people and that she doesn't know any better. They're too young to really understand though.
In my opinion, scolding doesn't really teach a dog anything. I mean, you can express your displeasure, but your timing has to be good, otherwise, she will not make the connection you want her to make. I mean, it has to be within seconds.
The best thing to do is to become a pro at managing situations so you don't put her in a situation where she will fail. And, it really helps to understand how dogs look at what we do. For instance, last night, you got her super excited. She jumped on your son, in my opinion, to play because she was so excited. So, if you want to exercise her and get her tired out before bed (GREAT idea!) then do it separately from the kids, OR, have her on a long leash. You can use a leash longer than her normal leash, that way you can keep hold of it, and step on it if she starts to run toward your son when she's hyper.
Also, when you yelled at her, as she ran toward him, it may have only excited her more, as to some dogs, yelling sounds like barking, which they do in play.
Lunging isn't always aggressive, it is done in play, too. And, time outs, in the cage can cause her to dislike the cage, so pick a different spot if your intent is punishment. If your intent is just to separate her from the children, the cage is fine.
Now, I'm not saying this situation was your fault, just that there are ways to manage the environment next time.
You can use the leash to control her, but also, in the meantime, work on recall (getting her to come when you call her), and teach some commands like "leave it".
Also, do a search for jumping. When she attempts to jump on people, have people turn their backs on her. Praise her when she has "four on the floor." Jumping when greeting people is a bit easier to deal with, though, than jumping on people when she's been playing. When she's greeting people you can get her to sit, a bit away from people, and let them approach her. If she gets up, you remove her, or simply back up, and try the sit again. Here's where having her on a leash inside can be a help!
My older dog used to pee when he was excited, too. He grew out of it!