The squeak when you hit a speed bump could be shock absorber bushings or sway bar/anti-roll bar link bushings.
The noise you describe as someone "beating a drum", does it only start making the noise when you have completed your stop and the engine is idling? If so with the engine running and the hand brake applied, lift the hood and see if you can sense whether the noise is coming from the engine compartment.and try to figure out what area of the engine compartment the noise is coming from. My guess is the cooling fan blades are hitting the fan shroud or something is fouling the fan blades.