Press Command and Space keys, type System Profiler and copy Model Identifier here. Early 2011 MacBook Pro supports AirPlay Mirroring
1. See if your Mac is compatible with Mountain Lion > http://www.apple.com/osx/specs
2. See if your apps are supported > http://www.roaringapps.com
3. Make a backup, open App Store and purchase OS X Mountain Lion
............I need to upgrade my OS to 10.8 in order to do that, correct?..............
No not exactly.
AirPlay Mirroring requires a second-generation Apple TV or later, OS X 10.8 or better and is supported on the following Mac models: iMac (Mid 2011 or newer), Mac mini (Mid 2011 or newer), MacBook Air (Mid 2011 or newer), and MacBook Pro (Early 2011 or newer). It also requires the computer to be using wi-fi.
However to do the things you have stated, you don't need mirroring, the Apple TV does all those things itself.
Model Identifier: MacBookPro7,1
Confused on "2. See if your apps are supported > http://www.roaringapps.com"
I don't use many apps unless they're in my Dashboard.
It looks like I can get Mountain Lion just fine.
So the AirPlay is for the iPhone and iPad and Mirroring is for the computers?
If all I need to do is get Mountain Lion, then buy the Apple TV (heard it's very easy to set-up) and starting "Mirroring", that seems easy enough.
Didn't see your phrase AirPlay Mirroring until after I hit send. Once Apple TV is installed on my computer the little icon pops up in the toolbar and I use that to control the HDTV/Apple TV interface?
Apple TV isn't installed on your computer. iTunes detects it on your network. The AirPlay icon you see is for basic AirPlay, not AirPlay mirroring, you can send content from iTunes to your Apple TV but not your desktop.
You don't need AirPlay at all to do the things you wrote hereinbefore, the Apple TV does them all.