Does $0 < x < 0$ imply $x =0$?

A false proposition implies any other proposition, so given the assumption on the left of 2),3), or 4), which are each false for any real $x,$ one could put any statement after "implies" in these and the overall statement would hold.

Here's a made-up example where a proof could use e.g. claim d) during the proof. The overall goal of the proof would be to show under certain hypotheses that $x=0.$ Suppose somehow the proof split into case A and case B, and that in treating case A one could show each of $x \le 0$ and $0 \le x.$ Here the use of a), namely $0 \le x \le 0$ implies $x=0,$ would suffice and finish case A in a mathematically sound way. On the other hand suppose in case B one could show each of $x<0$ and $0<x,$ thus arriving at the left side $0<x<0$ of claim d). Though it is logically correct to conclude here that again $x=0,$ since $0<x<0$ is false, in my opinion a better mathematical write-up of case B would be, once having arrived at the two statements $x<0$ and $0<x,$ just to say something like "thus case B cannot arise after all" or "so case B is contradictory".

I myself haven't seen a proof by a good mathematical expositor which used anything like claims b), c), or d) during the argument; as outlined in the above made-up proof scenario such proofs would just say such things as "this case does not arise" at the appropriate times.

$A\implies B$ means ($B$ or (not $A$)).

Let $A:0<x<0$ and $B:x=0$.

$A$ is false, therefore $not(A)$ is true. Thus $A\implies B$ is true.

See https://en.wikipedia.org/wiki/Vacuous_truth .

Please bear with me as a layman... but I'll try to explain why I was puzzled (and perhaps also why the OP was puzzled).

The other answers are correct if we assume that the natural language term "implies" used by the teacher as cited by the OP means the logical operation implication, or $\implies$.

That is probably what the teacher, tongue in cheek, indeed meant. What I, at first, understood though was that the first statement defines a set, and the second statement is about that set:

Let $\mathbb{M} = \{x | x \in \mathbb{R}, 0 \lt x \lt 0\}$; then $\mathbb{M} = \{0\}$.

Perhaps the proper notation would be

$\{x | x \in \mathbb{R}, 0 \lt x \lt 0\} = \{0\}$.

That's trivially wrong, hence the confusion.

What the professor meant was probably that the following statements are all true (given that $0\lt x \lt 0$ is always false, we can simply substitute $\bot$):

$\bot \implies x = 0$, or
$\bot \implies x \ne 0$, or
$\bot \implies y=42$,
or anything else.

(I'm not sure whether one can use $\bot$ that way; I may be too influenced by programming.)