I'm assuming you haven't played it-
You unlock animals by taming them, you get one at 50, two at 100, and three at 150.
So, I almost have the max number of snow rabbits.
The higher level your pet is, the more powerful it is, so having a level 27 is pretty good. It stops at 50.
And, badges are like ranks or levels, insect badge is one of the higher ranks.
In other words, I play this game a lot ;)