Yes.
> If so, then wouldn't an analog signal amp such as this simply amplify the noise along with the signal?
Yes. That's why there are limits on the reach of amplified networks.
> Decoding and re-encoding would have the effect of 'cleaning' the signal in this case.
Yes, but regenerating the signal introduces complexity, and thus reduces reliability, adds latency and only works for a specific encoding, which makes upgrades impossible. None of these things are wanted at the bottom of the sea for the next 25 years.
> If it does get noisy, where does the noise come from?
There are lots of noise sources. See: http://www.svphotonics.com/pub/pub029.pdf
> The medium?
Yes, partly.
> Leakage from outside the cable?
No. Although vibrations can cause noise.
EDIT There's a good discussion of what you're talking about here [0]
Upon further reading, there's a such thing as an all optical regenerator which would appear to provide the best of both worlds, though there's nothing here about how widely these are used [1]
[0] http://www.rfwireless-world.com/Terminology/Optical-Repeater...