I don't think there is a way to get a free site like mysite.com - the closest you can get would b mysite.hoster.com, like my site is darkblade.110mb.com.
110mb offers good reliable hosting with great support (no, I'm not working for them).
mbhosting is also good - look around for the best deal.
And if it is your site you can do anything you want (within the TOS of the hosting company) so yes, you can put a shoutbox on your site.
Hope this helps.