Special cases of the Laws of Exponents? I have a few questions regarding powers. 
We have:
\begin{align}
(x\cdot y)^n &= x^n\cdot y^n \tag 1\\
(x^m)^n &= x^{m\cdot n} \tag 2\\
x^m \cdot x^n &= x^{m+n} \tag 3
\end{align}
Question 1:
I guess a special case of $(1)$ is $y=x$, so:
\begin{align}
(x\cdot x)^n
\underbrace{=}_\text{$(1)$}
x^n\cdot x^n 
\underbrace{=}_\text{$(3)$}
x^{n+n}
=x^{2n}
\tag 4
\end{align}
Is this correct?
Question 2:
If a special case of $(3)$ is $m=n$ we have:
$$
x^n\cdot x^n
\underbrace{=}_\text{$(3)$}
x^{n+n}=x^{2n} \tag 5
$$
Is this correct?
But if $m=n$ in $(2)$ we have:
$$(x^n)^n 
\underbrace{=}_\text{$(2)$}
x^{n\cdot n} = x^{n^2} \tag 6
$$
And this doesn't make sense...
I guess it contradicts $(4)$ and $(5)$?
Thanks in advance!

I think your confusion may be the fact that you think you calculate the same expression three times, but in the first two cases you compute $\color{blue}{x^n \cdot x^n}$ in two ways (or using two properties) while in the last case, you compute $\color{purple}{(x^n)^n}$ and that's not the same:
$$\underbrace{\color{blue}{x^n \cdot x^n}}_{2\text{ factors}} = (x^n)^2 \color{red}{\;\mathbf{\ne}\;} \color{purple}{(x^n)^n} = \underbrace{x^n \cdot x^n\cdot \ldots\cdot x^n}_{n\text{ factors}} $$
