Is it possible to monitor web traffic data to and from an IP address knowing only the IP address?
Possible? Yeah. It's basically something called a "man in the middle" attack. Because packets aren't required to traverse the exact same links in getting from one location to another, you have to find some way to sort of "force" them to you.
Not being a lawyer, I think describing the details of such an effort are uncomfortably close to illegal activity, so I'll leave it at that.
I'll give you the benefit of a doubt that you want to do this on your own computer.
A network traffic monitor will log all this on your own computer and if you enable full packet logging you can look inside your outgoing packets to see if any personal or other sensitive information that you are sending is in plain text.
I do this to test connections with the various sites I have used over the years and with SSL my user name for that site is in plain text but the password and other stuff like emails and forms isn't are not in plain text.
I use IP Sniffer
All very good with their various nuances of logging and looking at your traffic for security purposes. It will take a while to get the hang of looking at your packet contents and what all that stuff is and means so be patient.