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I understand the setting as follows: For $n=0,\dots,N$, let $\X_n:=(X_{n,0},\dots,X_{n,k})$ and $\Y_n:=(Y_{n,0},\dots,Y_{n,k})$, where all the $X_{n,j}$'s and $Y_{n,j}$'s are iid $\mathcal C\mathcal N(\mathbf 0,\si^2\mathbf I_k)$.

The first problem is to find the distribution of
$$S_1:=\sum_{n=0}^N\X_n^H\X_n=\sum_{n=0}^N\sum_{j=0}^k|X_{n,j}|^2.
$$

As for your second question, the distribution of
For each pair $(n,j)$, we have $\frac2{\si^2}|X_{n,j}|^2\sim\chi^2_2$. So, $\frac2{\si^2}\,S_1\sim\chi^2_{2(k+1)(N+1)}=\text{Gamma}((k+1)(N+1),2)$. So,
$$\sum_{n=0}^N\X_n^H\X_n\sim\text{Gamma}((k+1)(N+1),\si^2),
$$
as you expected. If
$$M:=(k+1)(N+1)
$$
is large, then, by the central limit theorem,
$$\text{Gamma}((k+1)(N+1),\si^2)\approx\N(M\si^2,M\si^4).
$$

As for the distribution of
$$S_2:=\sum_{n=0}^N\X_n^H\Y_n=\sum_{n=0}^N\sum_{j=0}^k \overline{X_{n,j}}\,Y_{n,j},
$$
it is the $(k+1)(N+1)$-fold convolution of the distribution of the complex-valued random variable $\overline X\,Y=X_1Y_1+X_2Y_2+i(X_1Y_2-X_2Y_1)$, where $X:=X_1+iX_2$, $Y:=Y_1+iY_2$, and $X_1,X_2,Y_1,Y_2$ are iid $\mathcal N(0,\si^2/2)$.

In turn, the distribution of $\overline X\,Y$ can be obtained by the transformation-of-distributions technique (i.e., change of variables in a multifold integral; see e.g. Lecture 2) and is likely unremarkable. Mathematica worked several hours on getting the distribution of $\overline X\,Y$ and came up with nothing.

However, the mean and covariance matrix of the joint distribution of $(\Re(\overline X\,Y),\Im(\overline X\,Y))=(X_1Y_1+X_2Y_2,X_1Y_2-X_2Y_1)$ are $[0,0]^T$ and $\si^4 I_2/2$. So, if, again, $M$ is large, then, by the multivariate (here bivariate) central limit theorem, the joint distribution of $(\Re S_2,\Im S_2)$ is approximately the bivariate normal distribution
$$\N(0,0,M\si^4/2,M\si^4/2,0),
$$
with zero means, both variances equal $M\si^4/2$, and zero correlation.
That is, the real and imaginary parts of $\overline X\,Y$ are (i) zero-mean, (ii) each with variance $M\si^4/2$, (iii) jointly asymptotically normal, and (iv) asymptotically independent.