Newbie setup questions

Hi Couple of quick questions. I have just download teamviewer all I want to do is access my work computer via my laptop at home.

1. When I am connected and accessing the internet via my work computer does it use my home laptop ip address or my works computer ip address? Some of the websites I use monitor my Ip address that I have accessed the website with. It just causes me issues if I am accessing them from a different IP address.

2. Should I be buying some sort of licence? Or am ok with the free version? I'm only using it for access to my work computer? I'm not using it for commercial purpose. I.e. making money off using the application?

Thanks in advance! Chris