basically all film have a shelf life for reproducing optimum quality, an expired film may have fog, faded, or off color.
Yes, you able to push the film, depend on what film are you using, some films are not recommended for pushing, some films able to plus up to three stops, do note when you plus a roll of film, you are underexposed it and over developing it to get back the exposure, so you have to apply the same ISO thru out the entire roll, and tell the lab to push process it by how many stops.
Pushing the film will increase the contrast of the film and make the film very grainy, similar like you shoot underexposed on digital and bring up the exposure in photoshop.
and also pushing the film are cost more than normal processing charge.