With a TV that old, I doubt you will be able to read text on websites when displayed on the TV unless it is a pretty big font. I have a 52'' rear projection TV and it is hooked up to my computer as a 2nd monitor. While it is great for watching movies/videos, it is near impossible to read any text from documents or webpages. Sorry, just wanted to share that with you before you spend any money and end up disappointed afterward. Awhile back I shared your same goal and found out the hard way.
Unless it has an S-video connector I don't think it can be done. If you have an S-Video port you can purchase a RCA Video to S-Video cable. Can you give some more detail about the USB dongle and it features. Also can you give me the model of the eee box you have.