Hey there!
Well, I am not the expert on here...but I would say all of that is very very normal for this age. That was Brooklyn in a nut shell, and now she is much different (that happened through training classes and just general growing up). But even now, Brooky has the "crazy hour"
When she will run full speed from the living room down the hall outside circle the BBQ and back again. I have turned this into a game with her, doing commands, using toys, and sometimes just letting her go and burning off her crazies. All the things you just wrote below were all of my concerns too. I remember calling our trainer almost in tears because I thought she was crazy, hyperactive, she kept nipping us, she wouldnt listen. My trainer helped a lot to just accept this part of puppyhood, train and also, not expect so much so soon. Really, Brooklyn did not start "getting it" til about 6 months, those first few months were hard for me, I was so out of my element. But a good trainer and just enjoying the crazies is your best bet. I used to worry so much, or get concerned or annoyed or freaked out...and it turns out my energy fueled hers. Once I just chilled out and let it happen, focused on my training, Brooky was a changed dog. But again...that was much later. Most of this is just puppy...like it or not!
As far as games go, I think you just have to listen to your own dog. There are some toys (the Babble Ball is one!!) that just make Brooky crazy, like totally mental and hyper like you said. So, I usually don't play with them, or if I do, once I see her state of mind go too high, I just calmly take away and replace with a chew toy, bully stick or something calm. She is this way with bubbles too...she loves bubbles and it is calm and fun for a while, but I always have to stop before she gets the crazies (usually after 5 minutes), because they make her a bit nuts-o too ha. I mean, I play tug with Brooky, she is fine, we have fun. A good trick if you want her to release the toy (before you have taught "drop") is just to lightly blow on her nose. This instinctually releases their jaw...works every time
I also play a form of chase, I call it tag or hide and seek and Brooklyn loves it. But again, you know your dog so just pay attention to what is good for them, every dog is different. But those crazies/hypers you mention, really are just puppies. Sometimes they just need to burn off that burst of energy, no matter how long your walk just was! That is what my trainer told me. So I wouldn't say you messed up, don't worry
Maybe just play games that make you more comfortable about her energy level, but do expect those "bursts" of hyperness for a while.
With the biting. Totally normal too. They won't stop right away and you are doing the right things. When you screetch "ouch!" (or whatever you do to let her know it hurts even if it does not, screetch every time), then also at the same time, get up and walk away. Remove yourself from the game. She will soon learn she hurt you and lost a play buddy. Not something they want! If you stay around and keep playing, she won't get the message. End play. Also you can instantly replace with a chew toy (kong, bully, etc) if you want. But again, this won't stop right away even if you are doing the right thing (you are!!), it is puppy head, so just keep doing it. One day, you will go "ahhh! it clicked!!" Make sure Chloe has lots of chew things that are interesting too. Once she starts teething I used to put toys in water then in the freezer...cool chew toy and they love it!
So end all, you are doing a good job...and we are always here to help!! I learned SO much in my first year with Brooklyn, more than I ever thought I would have. And also remember that puppy behaviour is not always leading to aggressive behaviour at all. It is just puppy, and it is not your fault.
I gather, that age will cure 90% of this if you stick with what you know.
Also, are you set up for a puppy school yet? And then training classes after? Those worked wonders for us. Hope that helps! I am a rambler, sorry!!!