Yes, and remember that your Tamagotchi has to be a boy in order to obtain young kuchipatchi. But don't be disappointed if you don't get young kuchi because when your Tama evolves there is a bit of randomness.
Yes, Kuchipatchi comes from average care, so Young Kuchipatchi must come from average care, right? I'm assuming you should get a Mizutamatchi for a toddler, and then just start taking average care of your tamagotchi. Don't be shocked if your tamagotchi doesn't evolve into a Young Kuchipatchi, as GotchiGirl96 said, it is quite random.
Skill points causes it's growth also. I'm assuming you will get a few Intelligence points, a little tiny bit of Arts/Fashion points, and a lot of Social/Kindness points.