Bounds in NP-completeness proofs According to the polynomial-time-reduction definition

"If problem $Y$ can be reduced to problem $X$ in polynomial time, we denote this by $Y \leq_p X$."

If $X$ is one of already known NP-complete problem then we can say that $Y$ is NP-complete.

From my understanding, $p$ should be always polynomial and this cannot be an exponential function such as $2^n$ or $3^n$.

However, my question is if $Y$ or $X$ already has a lower bound that is not polynomial, which is $2^n$ or $3^n$ then can I still say that $Y$ is NP-complete?

In other words, does lower bounds for $X$ and $Y$ matter?
Note that you got it wrong in the question. What you have is that if $Y \leq^p_m X$ and $Y$ is NP-hard, then $X$ is also NP-hard. This is true by definition, unconditional on any other lower bounds known for $X$ or $Y$ (though, as Ryan points out, such lower bounds could be very interesting).

If $X$ is in NP and has a lower bound on its computational complexity that is super-polynomial, it would mean that $P \neq NP$.