Question: I have to buy a Raspberry Pi to control a monitor remotely. 

1. Can I install software like [TeamViewer](http://en.wikipedia.org/wiki/TeamViewer) on Raspberry Pi?
2. Can I connect the Raspberry Pi via wireless or LAN?
3. Can I connect an Internet key to a Raspberry Pi?

Answer: 
1. "*Can I install software like TeamViewer on Raspberry Pi?*"

Yes, use a [VNC](http://en.wikipedia.org/wiki/Virtual_Network_Computing) server.

2. "*Can I connect via wireless or LAN to the Raspberry Pi?*"

Yes, and you can even do both at the same time.

3. "*Can I connect an Internet key to a Raspberry Pi?*"

Yes, if you mean a USB Wi-Fi dongle.
TeamViewer can be installed on the Raspberry Pi using Eltechs Exagear. [Here](http://eltechs.com/run-teamviewer-on-raspberry-pi/) is description how to do it.