Is it possible, to extract the clean signal so it's just leaving the noise?
Strictly speaking, no: With sufficiently bad luck, your noise might just look like signal, and you loose information irrecoverably.
Practically speaking, maybe: However, if your SNR is good, you might be able to estimate the transmitted signal out of the received signal (which is exactly what a receiver does, it estimates the data that was sent!). Then, you can "re-synthesize" the clean signal as you received it, subtract it from what you actually received, and get the pure noise.
The spectrum of the noise is important for me to characterise. What would be the method? Cross correlation?
Well, the spectrum of the noise, assuming the noise is uncorrelated to the signal, will just be the difference between the received spectrum and the signal spectrum. If you know the signal's spectrum, you can estimate the noise spectrum simply by finding the difference between the two PSDs.
So, estimating the noise spectrum is easier than estimating the actual noise.