The check engine light stays on after oil change. When you apply the brakes the steering wheel vibrates and there is a weird humming type noise coming from the rear while idling.

Any wear or play in a bush or rod usually gives a 'knock' or causes vibration, not a humming noise.
If there's play in the steering wheel that may be due to a worn steering rack and pinion.

A worn ball joint can cause a 'crack' noise and depending on the wear, vibration that will be felt through the steering wheel. A worn ball joint can also cause the car to pull to one side.

A humming noise suggests it is either a brake rotor/pad, or a worn driveshaft or wheel bearing.

bring back to mechanic you definitely have some sort of oil pump or internal engine problem

built into the brake pads is a small bit of metal that when the pads get to low to be safe , runs on the rotor and produces the squeal applying the brakes stops the vibration so the noise drops off

Possibly a wheel bearing issue, does it `get progressively worse as speed increases or better as it decreases? does it stop when applying the brakes or get worse?

probably a bad wheel bearing

Inspect - Brakes, Rotors and brake pads.
Replace smart key battery, coolant,oil and filter.
Reset ( Maint Reqd ) light after oil change, if vehicle equipped with light.
Rotate tire's
Road test - While driving vehicle, check for proper operation of engine, transmission, brakes and steering also check for abnormal noise or vibration from any part of the vehicle.

check the idler arm on the steering drag link

check wheel bearing

my 325i makes a whining type noise when driving (not all the time), but stops when I apply my brakes. Any idea what this is?

Sounds like front differential or front drive axle issues. Check the front differential gear oil then take er' to the dealership before you end up spending a bunch of $. Most people change their oil frequently in the engine but forget about flushing the transmission fluid and changing oil in the front and rear differentials. I have three Ford 4X4s. Two which really get used hard. The 93 has 311K miles and still runs excellent. Trans and differentials get a fluid change every 25K miles and moble one synthetic engine oil changes every 5k miles.