I think the training points also play a role in waht character you get. And how many times you feed it snacks instead of playing games. Like for example, you can only get a mimitchi if you are on 2nd gen or higher, and at least one of your parents had a lot of training points. Then you also have to take good care of your tama to get the toddler kinkonomatchi (or whatever it's called) and THEN when it changes to a teenager, it has to be either young mimitchi or oniontchi. Again, good care and high disclipine is the key.
But what I've always wondered about three things:
1. how old does your tama have to be before it becomes an old timer?
2. Why is oyatchi an old timer, and not what you get if two old timers mate like in other versions?
3. How do you get a nazotchi?