if you're shooting an iso 1600 film, you can rate it at 1600, there is no pushing involved. some folks do rate it lower to overexpose by a 1/3 stop or so and to control the grain. it really depends what you're in for. if it's your first time shooting on the film, you may want to bracket the shots so that you can work out your preferred rating for the film.
whether is ISO 50, 200, 400, 800 or 1600 film, all will be go thru the same machine with same C41 chemical.
for color neg, you can overexpose a little to get more saturated colors and still able to retain highlight details,
if you underexpose it, you will get mute colors and losing shadow details.