There are a couple different types of connections that fall under the VPN umbrella. I’m not sure if they have proper names, since most people don’t differentiate, but I refer to them as “infrastructure” and “P2P”.
An infrastructure VPN is the more common type. You connect from the outside (WAN, internet, whatever you want to call it) to the remote router, and it routes you as if you are connected directly to the LAN. You are assigned an internal IP address, just as if you were physically plugged into the router with an ethernet cable. You can communicate with any device on the LAN on any port. You’re effectively “on the inside”, even though you might physically be on the other side of the country.
A P2P VPN is a little different. These are most often provided by a service like teamviewer or Logmein’s Hamachi. It will only give you access to the machines connected to the P2P VPN, not to the other machines on their local network(s). The IP addresses are provided by the service, and have nothing to do with your or your peers’ physical network addresses. It’s an entirely separate virtual subnet and neither your nor your peers’ routers are involved. The “router” for this network is the service itself, though all it really does is set up the connection. None of the data on the virtual subnet passes through the service - if tunnels directly from your machine to the machine of your peer.
How/where the traffic is tunneled depends on the type of VPN and how it was set up. Most use 80 or 8080, as those are the least restricted by ISPs and most likely to work. Some corporate VPNs probably use other ports for various reasons.
Your “route to the internet” can either be through your actual internet connection, or through the VPN connection, depending on how you have things set up (assuming the VPN router allows it). If it helps, any VPN connection works through a network connection that, to your computer, is indistinguishable from a hardware connection. It’s like you have two network cards - one connected to your local LAN, the other connected to the remote LAN. It’s just that the remote connection is entirely virtual instead of an actual chip connected to a physical cable. I honestly don’t know what the default behavior is for your software (despite using VPNs regularly), but you can almost certainly choose which you want to use for internet connectivity. For example, if your local address is 192.168.1.5, and your VPN address is 10.1.10.20, you should be able to tell your browser to use the VPN gateway (presumably 10.1.10.1) for internet access. Note that you have to do this in the software itself - attempting to tell windows to only use a virtual connection for internet will result in failure. I’ll let you figure out why 
As for comparison to a proxy, you can use the VPN connection for internet access and that would effectively be a proxy. But the main purpose is to get you inside a remote network, either for security purposes or to access ports that would normally be blocked to the outside world.
