To directly answer your questions:
1. How can I do this without contacting my ISP (and is it even
possible?; I don't care about paying for something that does this).
To get an IP allocated directly to your house, you'd NEED to contact the ISP. An alternative is to rent a VPN and then set it up on the server that you're running at home. It's more of a tunnel, but would be slower than direct allocation to your house, and likely 10+ times slower than a real VPS on real hardware and networks.
2. Once I have the extra external IP address, how can I forward it to
my VPS? How is my router supposed to differentiate between two
separate external IP addresses?
In your router configuration, there's usually a routing tables menu and you can directly point a static IP you purchase and point it to the LAN IP of some other server on your network. See [this document] for a similar question and answers.
Some other pointers and notes:
VPSs are very cheap depending on what
your needs are and where you look. Go with smaller companies (not too
small that it seems sketchy) or larger companies (not too huge that
they are doing it for mass-profit).
Your residential line probably can't support more than 15 megabits upload, which would be fine for running a few small websites but definitely not anything large.
Why would you virtualize (VPS) from your house if you have a "server" to run it on? Virtualization is only done to save resources and time in a real production environment when you need to isolate a large quantity of servers running on one machine. They're good in test environments too, because they're easy to reset if something gets messed up, and they don't use all the energy of a dedicated box.
I think you need to do more research, because from what it sounds, you're not fit to run a server out of your house.
Extended pointers and notes:
Your network can be easily attacked or invaded if you don't take the absolute proper security precautions
What I mean by that, is that someone can knock your internet down for as long as they choose, if they find your IP address. That's another reason to use a real server on a real network: they are more forgiving against attacks, and have better firewalls.
Just this week, someone was trying to gain root access to one of my servers (failed of course) but it just shows, that even when your IP is not being exposed (Not a web-server, strictly file storage) things can still happen. Once they gain access to your server, they can essentially control any and everything that happens on your network, such as host illegal files, attack other servers, capture your internet traffic and passwords, etc.
If you buy a static IP, there's really no need to "port-forward" since you can allocate all 65535 ports to it by setting it as a DMZ (De-Militarized-Zone). I mean, you could only open the ports you want if you have a firewall of sorts, but in most cases you'd want to leave everything open but set it up in a way that it's isolated from the other nodes on your network.
FURTHERMORE: Your ISP may not even allow you to host services from your home line. Check their Usage Policy first, or contact them.