Time out works with kids but not dogs. She probably thinks it`s a game.
The fact she will respond for food is excellent. This is something you can build on.
She sounds like a normal pup tbh.
I would get a good puppy book (or lots and compare them if you can) but NOTHING by anyone with a TV programme as they are just entertainers.

For a good book on dog behaviour, Jean Donaldson`s The Culture Clash is excellent.
Re your queries: establish a proper routine - same as you did when your children were little. Teach the dog two things only and reward every time: a Sit (in front). This can interrupt any unwanted behaviour. Please ask the children not to approach the dog unles with you. You need to train them as well!
Then teach Bed. Nicely! It`s not a punishment. Lure the dog to her blanket or whatever with a treat. Say Bed, give the reward. Build up till she will settle for a minute or two (don`t expect her to remain long - she`s a baby). This gives you a few seconds to collect your thoughts and for things to calm down.
For the nipping - this is normal. Dogs need to learn to control the pressure of the bite and this is how it is done. Every time she puts teeth on skin or clothes stop the game. Walk away. Ignore her. Eventually she`ll learn.
You need to go to a class - especially as she is coming up to the teenage stage. Look up APDT classes in your area.