Curves in the Web PKI are overwhelmingly NIST P-curves, which, again, are only deeply mistrusted on message boards, and when needed to get the BADA55 paper accepted.
New designs shouldn't use the P-curves, because it's too easy to implement them vulnerably (for all intents and purposes any random string is a workable Curve25519 point, and that's not the case for the P-curves --- you have to do fussy input validation). But that has nothing to do with conspiracy theories about how the curve was generated.
You don't have to take my word for this; you can just read the Koblitz and Menezes paper, which takes this question on in detail.
That the NSA picked the magic DES S-boxes in order to defend against differential cryptanalysis (which they'd discovered first), and that the NSA picked the Dual EC DRBG constants to backdoor it in a NOBUS type operation are both established facts.
Yes, curves in the Web PKI are largely P curves right now, but ed25519 is also standardized.
You're certainly free to trust the NIST curves, but my point still stands: there is no good reason not to pick such constants using a nothing up my sleeve algorithm, and the fact they didn't do that means they either know something we don't, they backdoored it, they wanted people to suspect they backdoored it, or they're idiots and decided to ignore established best practice for no reason. It's entirely possible the answer is the latter, of course :)
No, the logic in your last paragraph doesn't hold at all. You should read the Menezes paper rather than trying to derive this stuff from faulty axioms.
New designs shouldn't use the P-curves, because it's too easy to implement them vulnerably (for all intents and purposes any random string is a workable Curve25519 point, and that's not the case for the P-curves --- you have to do fussy input validation). But that has nothing to do with conspiracy theories about how the curve was generated.
You don't have to take my word for this; you can just read the Koblitz and Menezes paper, which takes this question on in detail.
So, no, really, none of this is true.