First case, we say that X and X are two different random variables. But I do not see any condition for X to be a random variable given that X is a random variable, so I think this statement is not true at all.

I repeat, the same method you use to construct random variables such as X², 1/X, and (X+3) also works to construct X.
Second, since we can not map back a real number to the sample space, say using X, then the fact that X is a random variable does not imply that X is a random variable too, so this statement is false.

What kind of thing is X? It makes absolutely no sense to speak about X unless you've given a meaning to X.