'When in doubt, sponge up more native Russian text and your brain will formulate these kind of abstract things for you.'
Yes, thankfully that tends to happen, which is great, except that I then have a less clear idea about the reasons why, which my pedantic side finds frustrating! Can't win with a brain like mine...

'Notice, that these are all basically just the noun верх, as an object of a preposition'
Hadn't explicitly thought of them on those terms, which is odd seeing as I made quite some effort to learn how those same prefixes applied to verbs.

наверх - upward (describing motion in an upward direction, ONto something, hence на)
вверх - upward (describing motion upward, into something, hence в. That something may just be the space above something else)

'In his correction, the man looks across over the papers to the guy.'
I see what you mean; I have a tiny suspicion that 'по' in this instance may be less directionally specific; that's just a vague hunch though. I find 'по' the most baffling of the prepositions. I think, bearing in mind what I know of (verbal) prefixes, наверх would have been a better choice than вверх , but I see вверх used more frequently, so was probably influenced by that.

"Up from the papers" in this case, is focusing on "up from below" rather than "into above". Perhaps снизу would be involved.'
I also thought that sounded as if he was looking up from the pile of papers while sitting on them! Which made me laugh. Did you mean 'сниз' here?
'подняв взгляд от вороха бумаг'
I had wondered whether to flesh out my original sentence to say more precisely 'he moved his gaze upwards from x to y', which is a construction I came across in 'Master and Margerita'. Judging from Serge_spb's reply, it may have been a better option. Also, 'ворох' is a better choice than 'множество'. I'll remember that word now.