In my example, I want to build an application that sends users who join a network some kind of interface and manage this at a central station (possibly the router, or a central server). The new user's input to this interface will be sent back to the central station and controlled.
How plausible is this? Is sending something to a newly discovered IP realistic?
As long as you control the DNS server, you can send them to any web server you like.
Completely plausible, but you'll need a router with open source firmware and you'll need to program in the language of that source code and have the toolchain to build the binary for the firmware.
The only thing I can think of is NoCatAuth and friends. The user has to use their web browser, but most are accustomed to that.
Are you trying to FORCE the users to use your application (e.g. by selling these routers via an ISP), or are you expecting users to co-operate (e.g. inside a organisation's WAN)?
If the latter, it may be sufficient to set the DHCP server inside the router to serve the address of an HTTP proxy. That will get picked up by most OS/browsers. The proxy can then be used to control web-traffic - which pages they can see, and which ones are redirected to your own web-app.
If the user is considered an adversary, it would be trivial for them to override the proxy settings. In a LAN/WAN situation, you need to make sure nothing is connecting them to the outside world, except through the proxy.