I'll summarize and paraphrase your questions for clarity:
Where does the light come from that illuminates Armstrong when he's climbing down?
The lunar surface. If you can see something, that means that light is reaching you. Some of it enters your eye, and the rest illuminates you. Armstrong could see the (bright!) lunar surface outside the shadow of the LEM, therefore he was lit.
If the LEM had been in a valley where no sunlit parts of the surface could be seen, THEN you'd have a case. If they'd landed in such a place, the only lightsources would be the stars and possibly the Earth, and darkness would be near-absolute. In this case, there is enough secondary sunlight to illuminate the rear of the LEM fairly well.

Why does he show up so bright, when the surface was in deep shadow?
This is where I attract the wrath of the board members by using the word "albedo". To put it another way: The lunar surface is a dark grey-brown for the most part. The astronaut suit is white. Makes sense that he'll show up brighter, no?

Why does he say that he has trouble seeing that he has good footing?
Because he does. There are quite a few brightly lit rocks in his vicinity, so his night vision is probably not all that good. Further, his feet and the surface he's standing on is less well lit than the rest of him, for the simple reason that they're lit from below. (see JayUtahs argument) And finally, the surface is a poor reflector, so most of the available light is absorbed. It's pretty freaking black.

What about the horizon remark?
I can't speak for the word choice - I go by my memory of what the descent video looked like. Not only the horizon is lit on the landing site, as is easily established from the photos.

Did I miss anything?

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