Really? Your uses can't see a BIG GREEN BUTTON that says VISIT THE SITE? It's been like that since version 1 from other devs... I'm the 3rd, if I'm not mistaken.
Regarding the domain name, I have absolutely no idea as I never used it. Will try to fix for the next version, that should be out in a couple of days or so.
And no, the link can't be there because there's a ton of ajax checking in that button to count the click, guests or not, etc. You can edit your template and add the link if you want.