Can a carotid bruit cause a persistent humming in my ear?

Yes: A bruit is basically the turbulence in an artery caused by a blockage. A good analogy would be when a nice wide silently flowing river narrows and gets very turbulent and loud like a rapid. If the humming you hear is pulsatile / cyclical then it could very well be coming from your carotid arteries. A simple carotid ultrasound can help evaluate the arteries to rule out any narrowing.

Yes: You ask good questions. If you have this then you need carotid doppler.