You can't do this without external hardware - there are no models of Pi with an HDMI or composite input.
There's an interesting writeup at dwellertech.blogspot.co.uk which suggests that it might be plausible to use a Hauppauge pass-through interface. They used an HD PVR, which is now hard to find, in the article but it's reasonably likely that the HD PVR 2 or the HD PVR Rocket might work. There are some drivers on Hauppauge's support page, but I'm unsure as to their viability for Raspbian.
- HDMI out from Sky box to HDMI in on interface
- HDMI out from interface to HDMI in on display
- USB from interface to USB on Pi
The HDMI signal's duplicated to the interface's output port for the display. A copy is compressed to H.264 and output to the interface's USB socket. The interface appears to the Pi as a video device (
/dev/video0), meaning that you should then be able to use a streaming package (VLC/FFmpeg/gstreamer/netcat/etc.) to transmit the stream to the network.
For the IR signals I'd suggest you look into lirc, and perhaps the python-lirc library. It should be entirely achievable to set up some infrared receivers and transmitters and shuttle the signals around your network.