This is an actual live chat provided by Xat, same rules apply here. Feel free to make a profile, if there's a few other people online then I suggest you check out the chat's page to have fun with other features like games or media.
This is just so it can have the chat retain it's information despite being embedded. If you don't wish to consent to it, go and click the chat page link to open it in a new tab.
Click your name on the right side, it'll pull up things to fill out. You can find the list of avatar numbers here if you wish to change it. Be sure to click save afterwards.
No, you can just leave your messages and run. I don't know how long your information will be preserved in your browser, but if you plan on being a regular chatter I would suggest making an account.