The reason for the two cables is that satellite channels are transmitted in two polarisations, and in two bands - so four different sets of channels, effecitvely, and you can only get one set down a cable. With two cables, you can guarantee that you can watch one channel and record another (and, if the other channels you want to record at the same time are in either of the two currently selected sets, you can record them, too, which is why you can sometimes, but not always, record more than one other channel). The Sky box sends signals up the cables to the LNB on the dish to tell it which set of channels it wants on which cable at any particular time.
send one cable to the new TV and leave the Sky box with just one cable, but then you'll only be able to record while watching another channel if the recorded channel is in the same band and polarisation as the one you're watching.
Yes - some platforms in other countries have limited their transponders so that they are all in just one of the 4 bands - meaning you can split the LNB feeds and feed multiple tuners from a single LNB feed. This only works if you have a small platform, but if you do this makes a lot of sense.
Some satellite dishes have a quad LNB (four connectors) with which you could "fully" (as in watch and record one other channel) serve two TVs/boxes or a distribution box (which has its own switching hardware to act like a satellite's LNB).
Though worth pointing out that Sky Q doesn't work like this in either of the two ways it can be installed.
Sky Q regular installs use a Wideband solution - so rather than the two LNB cables each carrying a feed for a single tuner (and each feed being switched between one of the 4 bands - HiH, LoH, HiV, LoV) - the two feeds carry either the Wideband horizontal LNB output or the Wideband Vertical LNB output (Hi and Lo bands carried together). This means that you can feed as many tuners as you like from the two cables - unlike the previous solution which would have required 4 cables from the LNB to feed unlimited tuners.
Sky Q also supports a Unicable II solution - which is far more complex (and requires the LNB to do a lot more processing).