In that cave you have to found the bones of the rabbits that the Ma... ahem, the bear ate.
Open the door at your left, dig the dirts and open the second door. Release the medium boulder to crush the sticks and the red rat. Open the left door (not the right, there's a blob behind it) and press the red button, it'll destroy the n°1 wall and allow you to get the bronze key.
Once you have the key open the bronze door and dig the dirts in the right order to get the silver key:
Open the silver door and press the red button n°2, it'll destroy the n°2 wall. Chop the sticks, mine the suspicious stones and take the bones. You'll also need the bones in the 'Wolf 's lair' cave to complete the quest.
You don't need any special tool to clear that mine.