The short answer is yes.
The long answer is .... sorta. Rooting your phone allows you (and your apps) greater access to the system functions and calls of the operating system.
With Apple iOS apps, they must go through an approval process before being published, with Android there is no approval process so most apps are already available in the Market. If an app requires root to function, it will be listed in the description.
Free Apps - Yes, you can get "hot" versions of apps but... No. Copyright infringement is not cool. Don't do it. If the app costs money, pay the money.