The way that the triple socket (FM/TV/Sat) works is that it is band frequency filtered. What that means is that although the FM, TV and Sat signals all occupy different frequency ranges, the distribution amp combines them and outputs all of them together down a single coax to each wall plate.
They remain distinct because they operate at different frequencies. The wall plate then has filters which pass or block certain frequencies. So the TV socket only passes signals in the 470-790MHz range.
FM............ 88-108MHz
DAB......... 217-230MHz (FM and DAB are usually combined as 88-230MHz)
Digital TV... 470-790MHz
SAT......... 950-2300MHz
This means that if you're trying to get a TV signal out of any socket other than "TV" then you're wasting your time. Ditto for FM/DAB and for Satellite if they are present at the amp end.
If you want a satellite signal from the LNB cable transporting from one room to another then you need either to send the signal in to the amp first (in your case, under the stairs), or you need a direct physical connection from one room to the other i.e. a bit of wire that does not go via the aerial amplifier/splitter. Both of these solutions require dedicated cabling. There's no getting away from this.
If you want to move the Sky box (or any Satellite receiver) to room 2, then really the best solution is just to get two new LNB feed cables off the dish direct to room 2. Simple. Job done.
The reason for two cables with Sky/Freesat is because Satellite works in 4 bands. A quad LNB with 4 outputs is really 4 LNBs in one. Each LNB is "tuned" to whichever band the particular channel is on. Changing channels often means changing the LNB band. Since the LNB can't be tuned to more than one band at a time, then watching one while recording another means the need for to LNB feeds to two individual tuners.
Single cable mode on a Sky box works by restricting the "other" channel choice to just those that are on the same band.
Freeview works differently. It's more like the FM/TV/Satellite combined signal. The various channel groups operate at different frequencies but are all received simultaneously by a single antenna and then relayed via a single coax to the TV tuner(s). A modern Freeview recorder has at least two TV tuners. This allows the "watch one, record another" feature.
If you are ditching Sky then either Freesat or Freeview is an option. Freeview is the simplest. Your existing aerial system will supply the signal required with no need for any additional cost. Invest in a decent box (Humax) and it's pretty much plug and play.
Freesat offers marginally better picture performance. The rub is the extra costs for cabling.