If your dog was 10 months I would agree that you have a serious issue.
Your pup is 10 WEEKS ... this isn't really a SERIOUS issue. He's a puppy, this is what puppies do. Now, I'm not saying what he is doing is acceptable, because it's not, but he sees your kids as his playmates. Have you ever watched puppies play? Especially GSD puppies? They are not all marshmallows and roses let me tell you. Kyleigh sounds like a baby lion when she's playing with her best buds (dog friends).
I haven't a clue how to link other threads in here, but there are TONS of threads, information, advice and suggestions in the puppy forum.
Every dog is different, so you'll need to weed out the ones that work for your puppy and the ones that don't. AND, it will take some time to figure out which one works and which one doesn't, and you always need a back up. Kyleigh figured out that if I did "a" she couldn't jump, so she'd try something else, and I needed to have "b" in my "back pocket so to speak" to let her know that THAT won't work either.
And I'm sure your wife is very overwhelmed, a puppy and two kids ... that's a LOT of work for one person all day. Do you have a crate for the puppy? There is absolutely nothing wrong with your wife putting the puppy in the crate for an hour or so while she gets the "important" things done, and then bring the puppy back out to play.
The time out area ... are you expecting a 10 week old puppy to sit / stay in an area for a long period of time? If you are, your expectations are WAY TOO high. Your puppy can likely sit / stay in one area and not move for about ohh ... 10 seconds ... and then his attention span is on to something else. This is NORMAL.
Please go through the puppy threads ... you'll discover a whole wealth of information.
And good luck, and have fun with your puppy! Before you know it he'll be a year old and you'll wonder where the time went!
Marionís Zoo-Kyleigh, Raylan-cat, Echo-TAG,