To respond to the last question first, I think the extra codes are designed to operate either alternative or legacy receivers as well. For example the same chipset in the Tx could interact with a very different Rx in a garage door or gate control using the 'other codes', without having to reprogram a whole new remotes. Just package up the chip with a new plastic coat and we may never realise that it is the same remote, unless it is analysed as you have done. What else does that company (or the chipset manufacturer) make that is RF controlled, I wonder?
I agree with everything above. Just to further explain (sorry about the longish post, a bad habit of mine being a now retired teacher) I agree the coding system for the switches appears simple enough and I am happy with your explanation of how 1 key can set the other 3. What I am suggesting is the 'curious' bits are a permanent (not rolling) unique factory number randomly chosen so it is very unlikely that a batch of remotes would ever have the same 'curious' bits and hence the same signature that is learned by the switches. (My weather station has an 8bit rolling code that is freshly chosen every time the power is lost, eg changing batteries, the sensor must then be re-registered with the base station, grrrr. But it does mean there is a 1/256 chance of another sensor belong to someone else interfering. This would not be unacceptable for controlling power switching in a house, but acceptable for low cost weather stations.)
However I think I am right in saying - someone above tested the switching by just sending the one pattern, and only sent it once, and it still worked? If that is true then I would guess there is some error checking code in the 'curious' bits. If not, then it is possible the same pattern has to be received say at least 3 times (out of the 4, or all 4) to be considered valid, in which case there would be little need for a checksum concept in the 'curious' bits.
Another thought is if the number (eg 1-4 switches) assigned to the signal the power switch (is apparently attached to the end of the bit packet) it would need to be incorporated into the preceding 'curious' bits if it was to act as an error checking system, otherwise it would not work.
Either way, if I was designing a receiver to listen to the remote I would just accept the whole pattern as a given, put any further diagnosis aside, program the Rx to learn the pattern, and then listen for at least 3 identical packets and use that as the validity check. Pretty crude, but I reckon it would be ok.
Cheers, Rob