I have quite a bit of experience remotely operating my station...
Actually you've chosen a pretty good way, although Teamviewer would not be my choice for remote access. If you are running the "pro" version of XP, 7 or 8, you can use Microsoft Remote Desktop (RDP). However to use RDP you need to be able to open a port on your router. If that sort of network work is not your thing, then maybe Teamviewer is the best choice for you. RDP is the lightest weight, speediest screen sharing method available, though.
As for rig control, as others have mentioned you have a choice between Flrig, Ham Radio Deluxe (new pay version 6, or the old free version 5--search for "HRD free") and DXCommander. You might also look into the Icom
RS-BA1 software. It costs money, but would work great with your 7200. RS-BA1 would get you SSB phone capability as well.
RS-BA1 would be the most capable. HRD the next most but it's very complex. I used to run HRD with my 7000 for remote access. I've never used Flrig, but I do use and like Fldigi very much. I also use DXCommander, but only in the background to automate log entries into DXKeeper.
Leaving aside the RS-BA1 solution, if you want to run SSB phone, you'll need to make a VOIP link to your system someway. A lot of people use Skype. Again that is a clunky way that avoids having to port forward your router, but suffers from high latency (time delay). If you do choose to do the necessary router configuration we can talk you through it, if you'll pardon the pun
Then you can use something like
Remaud which is a very lightweight, low latency solution.
Right now I'm running a Flex radio, so when I RDP into my station computer I see PowerSDR and have full control over my rig. For digital I use Fldigi. For phone I use Remaud. And since I have an Elecraft KPA500 amp and have it plugged into the station computer also, I can even run my amplifier remotely using the KPA500 Remote software. On poor internet connections I have to disable the panafall display otherwise I get problems with audio. Other than that it works very smoothly. On the remote side I'm just running RDP client and Remaud client, that's it. Everything else runs on the station computer. The one thing I need to get is an internet controlled powerstrip. That will help me avoid the always embarrassing but now thankfully rare call to the XYL, "Uh, honey, would you please go into the office and cycle power on the computer?"