Do I have to sterilize the baby bottles after every feeding?

No. Bottle sterilizing goes back decades to a time when people made their own formulas from unpasturized milk & mixed it with water from questionable sources. Certified city water supply eliminates the need to sterilize in most cases. A good bottle brush and the same attention you give your dishes should be adequate.
Yes. Yes - the infant transfers lots of bacteria to the bottles during each of the feedings. After the bottle has touched the infants mouth it is ok to re-feed off of it for roughly 1-2 hours. After that it needs to be fully cleaned.