plasmid

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1. How would you cross puzzle land? Part V

I believe this gets honors as the best "cheating" answer thus far.
2. The lion and the tamer, Part 2

These partial differential equations are beyond me at the moment. But for the follow-on question, I'll posit the following:

5. How would you cross puzzle land? Part V

That phrase was posted on Brainden?!? Regarding pumping, I'm not sure how that would work out in practice. If the starting situation has the acrobat on a platform with the trapeze rope essentially going horizontally, then if you add more force to the swing you'll end up going above horizontal on the other side before gravity pulls you back down, and you wouldn't be swinging like a pendulum but more like a weight on a rope that was just held out somewhere in space and dropped. So you could pump exactly once before letting go. And motorizing the pivot points of the trapezi is certainly enough of a cheat to qualify as a Puzzle Land answer.
6. How would you cross puzzle land? Part V

Correct, initial velocity is zero. The answer is independent of whatever value you choose for gravitational force, but you could use 9.8 m/s2 if you like.
7. How would you cross puzzle land? Part V

An acrobat can release the trapeze at any point and follow the usual laws of physics (of a point mass without wings or rocket engines or such) until coming into contact with the other trapeze bar. Will add a drawing next if that would help.
8. How would you cross puzzle land? Part V

The non-cheating solution I have doesn't involve the sorts of things you speak of. The acrobats can be considered as points that can attach to and release from a trapeze at a distance rope length from the trapeze's axis, and that just follow Newtonian mechanics aside from being able to grab onto the trapezi (or whatever the plural of trapeze is).
9. How would you cross puzzle land? Part V

I came up with a solution where the distance between the platforms could be greater than height * 4, even with a level floor. Having the platforms as far apart as you want and making the acrobats hoof it across the circus grounds counts as a legitimate cheating (and thus paradoxical) answer, and is so far the best for others to try to top.
10. I'm not a pirate

Right on, Wilson
11. I'm not a pirate

Not a hockey goalie, fisherman, or cartographer. I like how goalie fits with these clues very well even though the clues aren't interpreted in the way I had in mind. But the second stanza is difficult to fit with that answer. I like the reasoning behind the fisherman answer as well, but once the answer is revealed there will be a much clearer interpretation of the clue about Atlas and the last line's use of the word dearth implies that two would be an unacceptably low number (and for the sake of hinting, the dearth in the last line refers to Atlas' feet from the previous line). A cartographer is actually closer to my answer than you might think.
12. I'm not a pirate

Indeed not an asteroid impact. I'm thinking that once the answer is seen, all of the clues will become visible and make sense. (There might be hinting in here.)
13. I'm not a pirate

Not a blacksmith; while it could certainly fit parts of the riddle, the last stanza would seem to go unexplained. Also not a fencer, as I would consider that to be too similar to the decoy answer of not being a pirate. Not a lobster trap. I'm afraid I don't know enough about them to be able to judge how well it fits the clues, but that also of course means it's not what I was thinking of. Hint:
14. I'm not a pirate

Hi DD. Not a jigsaw; while in some ways it's on the right track, I'm looking for something with a more specific interpretation for the opposing sword and masking of thieving hand.
15. Fill the vacancy

I don't have a full solution, but this might help get things started. After that point, it gets too abstract for me to continue comprehending.