I'm very far from an expert, but it's an interesting question and I have a few thoughts.
Regarding the first question, "could a water world be stable for thousands of years with most of its surface remaining covered in water", being somewhat pedantic the Earth would meet this description, so yes (and I presume you really mean 'millions' if not 'billions' of years?) - although even then the "Snowball Earth" hypothesis would suggest that this state is not completely stable on those timescales. There is presumably a relatively narrow window of temperatures for stability, as too cold and the water will freeze to ice (e.g. the assorted ice-rich moons of the outer Solar System), while too hot and you'll lack a defined surface as water transitions to vapour. Both would appear to be runaway endpoints, i.e. as the planet heats up, the atmospheric water content increases, which increases the greenhouse effect, which promotes heating, etc.
Gravity might also be problematic, at low mass the atmosphere will be less dense and more easily lost, while at higher gravities the efficacy of water vapour as a greenhouse gas might lead to high temperatures and pressures. Which I guess is some kind of consideration for your second question.
I don't quite follow the tidal locking question, but this would suggest a reasonably compact orbit which would in turn lead to the temperature runaway and/or loss of atmosphere problem (and literal evaporation if there's no rock core). Maybe with a cool M-dwarf star this might be possible, I don't know.
A quick browse of the arXiv turned up an interesting looking paper from the journal Icarus
http://arxiv.org/abs/astro-ph/0308324
which might give some more insight (from someone who knows what they're talking about ;) but i've not yet had time to give it a read-through - skipping to the conclusions suggests that it is at least possible, if an ice-rich planet subsequently migrates inwards.
1) Yes, in the real world, only a tiny portion of the light scatters by the Rayleigh scattering. This may be reinterpreted as the simple fact that generic places in the blue sky are far less bright than the Sun. It means that the generic places of the sky become blue but the Sun itself remains white. For the same reason, distant mountains keep their color. Also, the distant mountains don't increase the amount of blue light from other directions much simply because the intensity of light reflected from distant mountains into our eyes is vastly smaller than the intensity of light coming directly (or just with Rayleigh scattering) from the Sun to our eyes. And even if it were not smaller, e.g. when the Sun is right below the horizon and the mountains are needed, we won't be able to easily distinguish that the blue sky actually depends on the mountains.
Rayleigh scattering is caused by particles much smaller than the wavelength, i.e. individual atoms and molecules, so it doesn't really matter which of them they are. The rate of Rayleigh scattering is therefore more or less proportional to the air density which means that a vast majority of it occurs in the troposphere, especially the part closer to the surface.
2) The changing atmospheric density only impacts the angle of the propagation of the sunlight substantially if the atmospheric density changes at distance scales comparable to the wavelength. If the length scale at which the density changes is much longer than that, the impact on the direction of light is negligible and calculable by Snell's law.
If you ever watch Formula 1 races, you may see some fuzzy waving water-like illusion near the hot asphalt. This is indeed caused by density fluctuations caused by the variable heat near the asphalt (well, a campfire could have been enough instead of Formula 1). However, in this case the direction of light only changes slightly because the regions of hot and cold air are still much longer than the wavelength (half a micron or so).
If you think about ways how to get density fluctuations comparable to the wavelength which is really short, you will see that the source is in statistical physics and the naturally fluctuating air density due to statistical physics is actually nothing else than an equivalent macroscopic description of the Rayleigh scattering! When you calculate the Rayleigh scattering, you may either add the effect of individual air molecules; or you may directly calculate with a distribution of many air molecules and the source of the effect is that their density isn't really constant but fluctuates. So these two calculations are really equivalent. They are the microscopic and macroscopic description of the same thing, like statistical physics and thermodynamics.
If the blue light manages to come from a direction that differs from the direction of the source of light, the Sun, then – assuming that the atmosphere doesn't emit blue light by itself, and it doesn't (at least not a detectable amount of it) – it is scattering by definition. To get a substantial change of the direction, you need small particles, and that's by definition Rayleigh scattering. So there's no other source of the blue sky than the Rayleigh scattering – although the Rayleigh scattering may be described in several ways (microscopic, macroscopic etc.).
Well, there's also the Mie scattering – from particles much larger than the wavelength, especially spherical ones, like water droplets. However, for the Mie scattering to be substantial, you need a substantial change of the index of refraction $n$ inside the spheres, which is OK for water. Also, the Mie scattering is much less frequency-dependent (because $n$ only slightly depends on the frequency, nothing like the fourth power here) than the Rayleigh scattering so it doesn't influence the overall color much. Not only during the sunset, some grey-vs-white strips on the clouds near the horizon are caused by the Mie scattering. The Rayleigh scattering really has a monopoly on the substantial change of the color.
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The color of the sky doesn't just depend on the atmospheric density, but also on aerosols and particulate matter. Mars has about 1% of Earth's atmospheric density, just as you specify in the question. But its sky looks pale orange during the day, whereas sunsets look bluish. So that's almost opposite to what we see on Earth. The reason for these colors is scattering mostly by fine-grained dust particles. A nice picture is linked here (NASA).
The more general question is if you can tell a planet's composition by looking at its atmosphere. But from the context it appears that you imagine standing on the planet and actually looking. In that case, you can tell from the example of our own planet that this is not enough to deduce the composition of the planet. On the other hand, spectroscopic studies of the atmosphere do indeed help pin down its composition, especially if combined with other measurements. Most importantly, you need a way to deduce the planet's mass. This can be done if you find some other object gravitationally interacting with the planet (either a moon, or a spacecraft, etc.).