The buzzing is NORMAL, as the cossie doesn't have a normal vacuum assisted servo, it generates fluid pressure to assist the brake function, the noise is the pump running.
It should normally run for around 20-25 seconds first turn of the key in the morning, coming on after just starting is normal, but it will tend only to do this when cold, once up to pressure you will rarely if ever hear it.
It just needs a little use to settle down that's all.
If the pedal is ok , then you have no air.
Any leaks will/should be spotted at mot, the shake sounds std for cars stored for this long, the discs rust, and will do this till cleaned up by the pad, so give the brakes some work to do, sometimes if bad enough they won't clean up so new disc time.
Track rod ends won't cause this unless totally shagged, but you'd feel that in other ways.
These things though should also be found in a any decent mot.
tabetha