I need to clarify that this is not how it works in HTM theory. We use an entire layer to represent location, a layer using SP as described in Locations in the Neocortex: A Theory of Sensorimotor Object Recognition Using Cortical Grid Cells. We are not encoding location using an encoder like you describe, it’s more complicated than that. I talked about it more in the HTM Hackers' Hangout - Mar 1, 2019 near the end of the stream. I suggest you try to read and understand the paper cited above.
Path integration is a better term than “origin independence”. It means once you learn how space works wrt your body, you can use that representation in all situations.
Re #3 above, I’ve talked to Jeff about this, using the term “map” instead of “sets of pairs”. Maybe listen to this podcast with Jeff: part 1 / part 2.