Cross-correlation of maximum length sequence with noise Background

I am using a maximum length sequence to find the impulse response of a linear system. I want to calculate the expected SNR. I know that my signal amplitude is scaled by the length of the sequence after cross-correlation, but I am not sure what happens to noise.

Idea

Since the sign of noise is random, the sign of an element in a maximum length sequence doesn't matter: [1,1,-1] may as well be [1,1,1]. Cross-correlation with a maximum length sequence then is like using a moving average filter except the coefficients are all 1's instead of 1/n where n is length of the filter. A moving average filter reduces noise power by the length of filter, and scaling by the length of the filter increases noise power by the length of the filter squared. Therefore the noise power increases by the length of the filter.

Questions

What is the cross-correlation of a maximum length sequence and thermal noise?

What is the cross-correlation of a maximum length sequence with a discrete sine (e.g. EMI from a clock signal)?
I assume that this is a discrete-time problem where the maximum-length sequence is a pseudorandom sequence $x[k]$ of $\pm 1$ values and the noise is a sequence $n[k]$ of independent identically distributed (iid) zero-mean random variables with variance $\sigma^2$.  Then, $\sum_{k=0}^{N-1} x[k]n[k]$ is also a sum of $N$ iid random variables and its variance is $N\sigma^2$, that is, the noise power increases in proportion to the length of the filter, exactly as you have conjectured.  So, the crosscorrelation of the maximum length sequence and the noise is a zero-mean random variable with variance $N\sigma^2$ where $N$ is the length of the sequence being correlated against. If the noise is modeled as Gaussian, then crosscorrelation value is also a Gaussian random variable.