I What false hints of new physics were most notable?

[Maximilian]

The first item on your list does not belong on the list! The reason is because any physics undergraduate with paper and pencil ca prove in five minutes than faster than light travel or communication is impossible.

This proves only that faster than light travel or communication is impossible according to existing, established theory. It cannot, even in principle, prove that the experiment was wrong. The question was about experiments which suggested new physics.

By the way, your links do not even establish that it is incompatible with established physics, but only with established metaphysics. So,
http://www.askamathematician.com/2012/07/q-how-does-instantaneous-communication-violate-causality/ writes:

"But, if you accept the basic tenet of relativity, that everything no matter how it’s moving is on equal footing, then one person’s instantaneous signal is merely traveling very fast to someone else, or even slightly backward in time to another someone else." But, sorry, this means "if you accept the metaphysics of the spacetime interpretation of SR and reject, for whatever reasons, the Lorentz ether interpretation of the same SR".

In special relativity, a particle moving FTL in one frame of reference will be traveling back in time in another. FTL travel or communication should therefore also give the possibility of traveling back in time or sending messages into the past. If such time travel is possible, you would be able to go back in time and change the course of history by killing your own grandfather.

No. It would give this possibility only if there is no preferred frame. The FTL communication would have a preferred frame, and therefore violate the equivalence principle. But so what? In the strong form - that there is no real difference - it is a metaphysical belief. In the weaker, physical form there is no observable effect which allows to distinguish them. But new physics could give new observable effects which would allow to distinguish them. The FTL communication would be simply new physics, and give us an observational possibility to identify a preferred frame which is now hidden from observation.

There is a difference between a statement being factually wrong, and a statement being meaningless. The statement that the Great Pyramid of Giza has a volume of three cubic inches is factually wrong. The statement that the Great Pyramid of Giza has a volume of -45 cubic feet is meaningless because the volume of an object can not be a negative number. To claim that the velocity of a neutrino could be faster than light is a similar meaningless statement.

The only meaningless statement would be a combination of "there exists FTL communication" and "there is nothing violating Lorentz symmetry".

So, the superluminal neutrinos have their place in the list.