What you're referring to is usually called the chainring. Technically both the rear cog and chainring are sprockets.
Yes you can keep the same chainring if you buy a new cog with less teeth.
There are two common standards today when it comes to sprockets and the chain though. The more road standard, 3/32, and the more track standard, 1/8. This just refers to the width of the chain in inches, be sure your drive chain works together before you buy anything.