Glad I could help.
Sorry about this long rant. I don't know the exact answers to all of your questions, so I'll give some speculation for those questions.
You are on the right track. One detail to keep in mind is that it treats predicted inputs differently from non-predicted inputs in a few ways. I'm not completely sure, but based on the link below, it seems like the inputs to the HTM need to come in a known sequence for it to pool anything. So if the input suddenly jumps to the middle of a known sequence, it won't understand what the sequence is, at least not right away.
I'd like to add a link to sheiser1's list which I've found helpful: https://github.com/numenta/nupic.research/wiki/Overview-of-the-Temporal-Pooler
I'm not sure, but I suspect the TP can store a lot of sequences, similar to how the SP can represent an enormous number of things using the properties of SDRs. I'm not sure if the TP produces representations with those same properties though, but it probably should if it doesn't. Then again, there are way more possible sequences than inputs, so it might not be enough capacity, especially for very long sequences.
I'm not sure how much the TP could help long term predictions. It would probably filter out at least a decent portion of the instability to allow abstraction, so high regions might not be able to contribute to low level predictions very effectively, but they could still make abstracted predictions about the future. The final version of hierarchy will probably involve something more complicated than TP because there are multiple layers with many mysteries, and there are multiple pathways both up and down the hierarchy.
A predictions from predictions system might be able to work very well, but there could be technical issues. For example, you might need to distinguish cells by how many steps they are predicting into the future so that each prediction leads to the next prediction rather than predicting another step into the future based on all predictions of activity far into the future. I'm not sure if that could happen in biology.
I'm not sure the brain even makes predictions far into the future, at least of the same type of the temporal memory. Thinking about the far future probably usually involves things like imagination, extreme abstraction, and activity modes such as oscillation frequency. Longish term predictions could be useful for behavior in a simple reward learning system, but after a certain distance into the future of prediction, it probably needs to get a lot more complicated because its probably really hard to keep track of the long term impact of each action, and the number of possible sequences is too large to even experience many of them so some way to generalize experiences to concepts is required.
That said, there's probably at least some sort of medium term prediction - not on the order of milliseconds or seconds like lower cortical levels, but also not on the order of days like people are capable of planning. The hippocampus is capable of predictively playing sequences which it will later play as the animal takes a path. It can also learn to play a sequence if it needs to keep something in memory relevant to behavior, such as red flag means take a left when you go into the maze and blue flag means take a right, although I'm not sure how long animals keep the sequence going. That's related to the idea of long term predictions because it knows what it needs to do long before it actually acts.