My biggest question is, what do repeated interval jumps of perfect fifths have to do with the business of naming keys and the order in which notes are sharped/flatted?
Recall the pattern of whole steps (W) and half steps (H) that make up a major scale: W-W-H-W-W-W-H.
That creates the following pattern of scale degrees, which of course repeats in each octave:
W W H W W W H
1 2 3 4 5 6 7 1
Now look what happens when you create a new scale (below) that starts on scale degree 5 of your old scale (above):
W W H W W W H W W H W
1 2 3 4 5 6 7 1 2 3 4 5
W W H W W W H
1 2 3 4 5 6 7 1
Notice how 5 in the old scale becomes 1 in the new scale. Similarly, 6 becomes 2, 7 becomes 3, and so on. However, note that 4 does not quite line up with 7. This means we have to raise scale degree 4 in order to get the new scale's degree 7. By 'raise' I mean that if it is just a natural note, we sharpen it, but if it is already flat, we have to naturalize it (this will be important when I answer 3 in just a moment).
So to summarize, in order to create a new scale on the fifth degree of the current scale (i.e the key to the right of the current key), we have to raise (sharpen or naturalize) the fourth degree of the current scale (i.e. the note to the left of the current key). This explains the pattern on the sharp side. For example, to go from D to A (to the right = up a fifth), you have to raise the fourth of D, which is G (to the left = up a fourth).
Why is it key of F, but not key of F#? Why key of G, not G#? I would expect it to be symmetric to the flats
Why does this pattern span two keys in the sharp side, but only 1 in the flat side?
These questions both deal with the difference between the sharp and the flat side, and I think the answer to them is the same. The reason is that your final row changes definition partway through. On the flat side, you are saying how the key is different from the one to its right. On the sharp side, you are saying how the key is different from the one to its left. At C itself, you aren't comparing to either side -- that's why there's the gap. If you remain consistent with one or the other, the discrepancy goes away, and the pattern become uniform.
For example, to stick with the sharp side definition, we've already seen why G is like C (one note to its left) but with a raised F (two notes to the left). But you could also say that C is like F, but with a raised B♭. In this case, 'raised' means natural. Similarly, you could say that the key of B♭ is like E&Flat; but with a raised (naturalized) A♭.
I'll let you work out going the other direction, but I'll point out that in this case we are lowering notes (either flatting naturals, or naturalizing sharps) instead of raising them. So the key of C could be considered like the key of G, but with a lowered F♯.
What happens to this pattern in enharmonic equivalent keys past 4 #/b? Key of Db/C#, Gb/F#, Cb/B.
You could very well continue this pattern indefinitely, moving to double sharps after you've exhausted all the single sharps, and so on. There is such a concept as the infinite line of fifths. However, at some point, it will probably become convenient to wrap around to the flat side via enharmonic equivalence. When this occurs, each note can be thought of as occupying two spots along the line: one on the right side, and one on the left. They are just two different names for the same relationship. But the same pattern will hold for both (remember, once you rewrite your final line to be consistent as above, it will be the same uniform pattern on both sides).