Why are push-propellers so rare, yet they are still around? Jet engines are by their very nature push-engines, however, most propeller airplanes use pull-engines. Is there an inherent advantage to using pull-propellers except for the increased airflow over the fuselage and tail (with its rudder and elevator)?

Twins generally have their engines on the wings, and the tail is no longer directly behind it, does that mean the choice of a pull-engine is not as advantageous?

If there isn't an inherent disadvantage, why are pusher configurations so rare? If there is one, why do they exist at all? Disregarding designs where the choice is obvious, like powered parachutes where you simply don't want a propeller in your face.

The Convair B36 is one notable multi-engine aircraft with engines in pusher configuration, as is the Piaggio Avanti. And the Cessna Skymaster is a push/pull configuration (If you get a multi-engine rating in a Skymaster, your ticket will be limited to multi-engine aircraft with in-line thrust). Single engine aircraft are even more uncommon, and pretty much all I could find except the Lake Buccaneer are all kit-planes (e.g. Velocity, Rutan), ultralights (Quad City), military, or experimental.
There are many disadvantages, they seem to outweigh the advantages.  

Here are two:


A pusher prop is working in a disturbed airflow, causing increased vibration and noise
If the propeller is fitted behind a wing, each propeller blade is passing through the separated boundary flow twice each rotation. These cycles create additional noise and lower the efficiency of the propeller. The vibration makes the propeller blades more susceptible to metal fatigue. 
Propeller clearance at takeoff
Due to the pitch up at take-off, the propeller gets close to the ground. Therefore the diameter needs to be reduced (loss of efficiency) or the landing gear struts need to be made longer (added weight). Since the propeller is behind the landing gear,  it is susceptible to debris kicked up from the gear, increasing the need for added blade protection (increased weight, loss of efficiency)


Wikipedia has a list of additional disadvantages. 



Edit: 

Your statement that 


  Jet engines are by their very nature push-engines


is not entirely true.

In turbofan engines, most of the thrust is generated by the fan and compressor stages. Even in a pure jet engine, a lot of thrust is generated by the compressor. Therefore the shaft of a jet engine is tension loaded, just like a propeller shaft in puller configuration. 


Source: Rolls Royce - The Jet Engine