HDBaseT will very simply accomplish everything you're asking for.
Too bad it's only just now making a breakthrough, and stuck on 100Mb network speeds for now, and no one has Valens's chip installed yet.
But soon!
As for using a single remote for every room in your place, what would be the point? You're going to carry the remote into every room with you? Why not use the remote that goes to the TV in that room?
To do this as cheaply as possible
-- Stuff you're going to need --
HTPC (obviously)
RJ45 splitter
Spool of Cat6 cable - Be sure to get solid core, do not get the copper clad aluminum ( This link is just an example.)
USB over Ethernet extender 1 set per each tv you're going to
HDMI over Ethernet extender 1 set per each tv you're going to
FLIRC 1 for each tv you're going to
USB Hub
-- Setup --
Run usb and video/audio to each room via ethernet cable ( 3 strands of ethernet cable)
Have a FLIRC in each room to pick up the remote signal and send the commands back to your single machine.
-- Note --
You can go with an HDMI splitter since you will need a set of extenders for each room anyway, but going with RJ45 splitters will allow you to only have to use 1 HDMI extender over Ethernet at the source, instead of having 1 for each room. Plus they're cheaper.