Yeah Chromecast is about the best, some TVs have it built in as mirror too.
iOS or Android? There is some software but I can't remember, I'll ask around at work tomorrow. (I'm a mobile Platform Product Manager so should be able to fine out if you don't get a response from someone else on here)
Android. I believe it relies on Miracast . So you have have Miracast on both Phone and PC then if you have a wireless adapter that supports WiFi direct. That creates a peer to peer connection between the phone and PC hence no need for a router however I'm not sure if this just allows you to transfer files. I want to be as to mirror/cast the phone onto the pc monitor.