I only signed up a couple of days ago, but my understanding is that...
a) Broadcast shows are generally available for about a week afterwards as videos
b) To run to a TV, it's easiest to use Chromecast. This is a dongle (£30) that plugs into the TV HDMI port and receives pictures sent by WiFi from your laptop or whatever. There was an app for smart TVs, but apparently that's stopped.
c) I think it tunes the picture resolution to what your broadband will cope with. In my case, it's noticeably worse for the first 4 or 5 seconds than it generally is, and if I check the resolution in the settings, I find it's set to "auto".
For me, currently,
http://www.speedtest.net/ is showing 33 Mbps, and Eurosport player is offering up to 720p resolution in the settings.