1. A static IP is actually not necessary, but what you need is a consistent identifier. For the server, that's typically a DNS address, but for clients and peer to peer networks there's other ways to identify devices, usually tied to an account or some other key kept on the device.
2. For centralised communications yes, you would need an always online server. For decentralised networks, you just need a sufficient amount of online peers, but each individual peer does not need to be always online.
3. Pretty much, yes. Even push notifications on cell phones work this way.
4. Route, yes. Manually. VPN is usually not necessary. In modern web-based services this is typically done with websockets, which are client-initiated (so the client address can change), and which allow two-way communication and typically only require a keepalive packet from the client every minute or so.

There's other reasons why universal addressing is not done - privacy, network segmentation, resiliency, security, etc. And while IPv6 proponents do like to claim that local networks wouldn't be strictly necessary (which is technically true), local networks will still be wanted by many. Tying this back to phone numbers - phone numbers work because there's an implicit trust in the telcos, and conversely there's built in central control. It also helps that it's only a very domain specific implementation - phone communication specifications don't change very often. On computer networks, a lot of work has been done to reduce the reliance on a central trust authority. Nowadays, DNS and SSL registries are pretty much the last bastion of such an authority, with a lot of research and work having gone into being able to safely communicate through untrusted layers: GPG, TOR, IPFS, TLS, etc.