While the chromecast's inability to work with captive portals is it's own problem, they make tiny wifi routers, half as tall and twice as thick as a smartphone, and they're cheap. If you find yourself facing captive portals more than once, I'd recommend buying one so you can use your Chromecast in hotel rooms, among other use cases.
It shouldn't. Obviously if you're casting from a WAN source (like YouTube or Netflix), it needs WAN access, but if you're casting from a source on your local network, it should work without the WAN.
At least the upcoming Chromecast with Google TV comes with a remote, so you should be able to interact with the captive portal and use it that way.