Hey Zef, you can rename a studio and a pickaxe both within the Studio Manager.
To rename a studio, just click the Edit Page button in the top right and then enter a new name. Here is a post explaining how to rename a studio in more detail.
To rename a pickaxe or change the description you follow a similar process. Just navigate to the page of your studio with the Pickaxe on it, then click Edit Page button and enter a new description. Be sure to save changes.