Who told you that when you chat with someone on Omegle or other random website you will be anonymous?
Well, I developed my own Omegle kind of website and the first problem I had is the heavy load of video/audio streaming, so I had to a P2P solution which connect the users directly to each other and start streaming video/audio.
apparently Omegle are using an old method of video chatting similar to chatroulette.
An example of a map generated using these data automatically is http://www.chatroulettemap.com/
what can anybody do with your IP address ?
How does it work, how can someone get my up through Omegle?
The stream of your camera is not hosted through Omegle, it's generic TCP connection between you and the other side using a flash app that is actually cashed on your and their machine, and and ip can be simply found using netstat -a DOS command.
This is a huge breach to your privacy so the only choice would be omegling though a VPN connection, a bot that clicks next and capture ip and picture of each user is possible and does exist, and I wonder how much info you can get from just the ip address.
Location and exact address from an IP address, it's called geo2ip.
Contact info and phone number from an ip address "depends on isp" it's called whois.