Is it permissible to use the Bitly website, which displays potentially inappropriate ads in exchange for money, especially since it features a bee (a living creature) as its logo, and is the money earned from it considered lawful?

It is permissible to benefit from the website in what is lawful, and one must lower the gaze from what is forbidden. If the intention behind placing links is to earn profit from advertisements, then these links must be controlled to display only what is permissible. Otherwise, it is not permissible due to the potential for aiding in sin by promoting forbidden things. Justifying it by the actions of someone thought to be good is not sufficient.