This actually sounds pretty fun, although I know I'd waste a ton of money and make it way more work than necessary...
I haven't read through the Viking manual, but I know with Doorking, there are multiple modes you can use - some work with standalone phone lines and dial numbers; some work as hotline extensions, and some will route your own POTS line through them and can ring themselves through distinctive ring. It's all doable through the programming. But as I said, I didn't read the Viking manual to see how it compares.
Asterisk alone isn't the issue - there are plenty of simple systems like RasPBX that have the brains - but you need something that can interface with the POTS lines - either FXO or FXS depending on the door station - to deal with the analog interfaces.
If it were me, I'd ditch the analog lines - or at least interface both into a PBX with a cheapo gateway, then go all VOIP in the home. I'd find an interface to the analog lines to mix them both in - 1 FXO and one FXS (look @ linksys 3102) then deliver both to a VOIP system. But admittedly, that creates a lot more potential failure points, expense, and complexity... but that's just how some of us do things!