Level 13 Rewards, Guardians of Azuma
Raising Kai and Kurama to Level 12 gave me Life-sized Cutouts of them. All Profile characters, apart from your own child, will give you their cutouts on reaching Level 12.
I am almost at the point where the few remaining characters will reach Level 12 so will confirm this for every one.
Kai reached Level 13 this morning. He gave me a Hellfire Dango. This is one of the dishes you and he make together.
Kurama reaching Level 13 gave Victory Stones.
Although my son Misha gave me no item when he reached Level 12, he did give me his Cheery Castella on reaching Level 13. So yes, they each will give you one of the special dishes they make with you on reaching Level 13.
Murasame gave me a Deployment Set upon reaching Level 13.
Then, when they reach Level 18, they give you the other special dish they make with you.
At Level 18, Misha gave me the Split Mix.
Now when Kai reached Level 23, he gave me Hellfire Dango again.