Regarding your question about vibration.....it would help to know where exactly the vibration is coming from. If you feel it directly under you then I would look at your driveshaft/driveline. Specifically, look at the CV joints or center bearings. These tend to wear down in 525i's and unfortunately when they do you have to replace the entire shaftline which can cost about $500 reconditioned or $700 new (not including labor). As an alternative, check your hand-brake. Sometimes the hand-brake mechanism deteriorates and causes the underbelly to shake.

Regarding the gas pedal starting issue....check one of your fuel line hoses. Sounds like there's a fuel dispersion issue.

Regarding the red-light...this is a security/alarm thing. It usually goes on when the car is locked. No need to worry about the car battery being drained.