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First reading of the clues made me panic, but after a slow start the grid fill wasn't as bad as first feared. As for the origami, I haven't a clue about any of it!!!!

There is no origami; it is just rearrangment of parts. The first thing is to find the theme, and one of the clue answers gives you a hint. That should lead to a PDM and then the rest is easier than it looks.

The clues are mostly pretty straightforward, meaning that nearly all of the puzzle is in the endgame. Which is poor balance.
A few years ago we had a lot of puzzles requiring cutting up. I didn't see the point then, still don't...

My first “cutting up” puzzle and I really enjoyed it. The theme came to me in a flash - perhaps as it did to our hero.

The theme came to me by looking for possibilities of the 5,5 and 4,8 in the grid. They are both there in straight lines. The 4,8 was easier to spot, being a whole column.

Thanks Meursault - having completed the grid in a couple of hours I must have spent at least as long staring blankly at the grid. I can now see both names you allude to, though what on earth we are to cut out and where I still haven't a clue!

After a run of what I would deem to be 'sensible' Listeners it seems we are back to daft and ambiguous ones, a shame as I fear it will lead to me giving up doing the Listener and getting my Saturdays back.

I'm puzzled by "encompass". Does this mean that, for example, the hero is entirely within his 7-gon, or that he forms part of the boundary? I'm assuming, since 7-gon 2 is a reflection of 7-gon 1, that they are congruent, each of 72 cells.

I haven't pursued it any further, but I think you're very likely correct, Cockie, on both counts. A) Congruent B) Using all the cells of the grid.
If I was going to take it further, I'd invest a little effort in seeing how 1, 13 and 36 could be brought together...

Is one edge of the heptagonal shape diagonal?

On that subject, note that you cannot set off on a journey that includes 7 right-angles and expect to end up where you started. So it looks as if the scissor-work is going to become too tedious to bother with.