It is possible to push-start a manual transmission Contour with the following procedure:
1)Put key to ON position (check engine light should be lit).
2)Get a helper to push the car and get it moving.
3)Select 3rd gear.
4)Release the clutch for a second--engine should start.
You might be able to do this in reverse with the car rolling backwards, but reverse is a lower gear than 3rd and so you may find that it won't work.
As far as the problem with the starter, it sounds like you need a new starter.
