1 – the version is webtrees 1.5.3 (should have been upgraded a long time ago!). You can always see the version by pointing to / hovering your mouse over the webtrees logo in the footer. Same for kiwitrees.
2 – There is nothing custom about that page apart from the “Click here to visit my Blog” link in the header. The rest is entirely standard for the ‘xenea’ theme.What you are looking at is the “Home page” block. No different there to the way it has been since PGV days. IMPORTANT – that is the view of the Home page when not-logged in – don’t confuse it with either the same page when you are logged in, or My Page. Both display different links, for obvious reasons.
EDIT – just remembered – you also have to have “Allow visitors to request account registration” set to ‘yes’ [In admin -> Site configuration] to see a “Request for new account” link.