Why is my car shaking when stopped and smelling like burnt oil?

I'd suggest getting the Idle Air Control Valve checked. On most modern engines, the IACV can be adjusted yourself with basic tools. Some are adjusted by simply turning a screw, whereas some you may have to unscrew a small unit, and twist the inside spring. The IACV adjusts how much air is mixed with the fuel while the engine is idling. Not enough air will cause the engine to shake, as it struggles to keep going; while too much air will cause the engine to over-rev, or "breathe" (alternating high and low revving).