No, there are various other mechanisms now, perhaps the most useful and widespread for this situation being WebSocket. It's not without limitations though: http://en.wikipedia.org/wiki/WebSocket
Not really a proxy - it's pppd in Linux with a vty redirected to a socket. More like a cloud-based dialup ISP that tunnels over existing connections than a proxy.
And it's not quite "the browser" in the usual sense, but Chrome Apps (and, I believe, extensions?) get access to TCP and UDP APIs: https://developer.chrome.com/apps/app_network