Your Intelligence or Charisma score increases by 2.

Your tiny size makes you harder to hit and more nimble. +1 dexterity increase

The frog is part of the inside of the hat, where time has stopped. Your frog body still requires food but can 'age' indefinately 

Hats usually take on the traits of their creator but are also sentient and can go the opposite way out of spite.

Literally a frog, your size is tiny

Your base hopping speed is 25 feet. You have a swim speed of 25 feet. You have a climb speed of 20ft

You are proficient in one INT skill, and a toolset of your choice

You can speak, read, and write Common, Aquan, and one other language of your choice. 

By absorbing oxygen through your skin, you can breathe underwater, provided the water can supply the oxygen. You are also proficient in stealth due to your size. You can also use your tongue to grab and pull things from up to 5ft away that weigh no more than 5lbs, such as bugs, levers, or enemies' eyeballs.

When the frog controls its body, it is still under the hat's influence and won't do anything that attempts to remove it or generally things to get it killed, like swimming with crocodiles. The frog may croak, eat bugs or relax in a pond. However when the hat is in control (no check required) the frog's eyes will turn black, and it gains the ability to speak, and the ability to cast spells.