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Is it (the output, i.e, spherical, flat, hyperbolic) describing the physical shape of the universe and its curvature at any given instance of time or is it describing the shape of the expansion function applied to the universe? (i.e, the evolution of the universe spatially over time)

So a cone shaped universe could have a "flat" solution applied to it and this just means it keeps expanding forever. Or a "positive" solution applied which means expansion will eventually halt and reverse, ending in a Big Crunch. The original arbitrary shape (conical) not being affected at all?

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To answer your question it's important to understand exactly what the FRW solution is. The GR field equations give you a way to calculate the metric given some distribution of matter and energy (aka the stress-energy tensor). If you assume the universe is full of matter that is homogeneous, isotropic and non-interacting (except for gravity), then feed this into the field equations they tell you that the metric describing this universe is the FRW metric.

The metric allows you to calculate the curvature of the universe locally i.e. if you take some point you can work out the curvature of the universe around you. It doesn't tell you global properties like the topology, so for example the FRW metric wouldn't tell you whether the universe is an infinite sheet or a torus. Both would be possible.

All we get from GR is the metric, but then this is all we need. The metric tells us the local curvature, and because we started with the assumption that the universe is homogeneous and isotropic we know that the curvature is the same everywhere. Note that the metric describes spacetime not just space. If you know the metric at some starting point you can use it to work out the evolution of the universe in time as well as space. In fact this is exactly how the big bang theory came to exist. The FRW metric predicts the universe must have started at a singularity, and that it will end either as another singularity or infinite expansion depending on the value of $\Omega$.

So to get back to your question, there is no "shape" that is distinct from the metric. The metric/curvature is all there is. I mentioned that the global topology could be anything, but I suspect this isn't what you meant by "shape". Incidentally, when I mention the topology could be a torus, that doesn't mean space is curved into a dougnut shape. The topology just describes the connectivity i.e. a torus just means that whatever direction you move in you'll eventually get back to where you started. The universe could be a torus and still be flat (I'm not sure if an $\Omega$ > 1 universe could be a torus).

Were you thinking about the topology when you mentioned a cone? The problem with a cone topology is that spacetime wouldn't be isotropic. There would be some directions (around the cone) where you get back to where you started, but other directions (away from the tip) where you wouldn't get back to where you started. Also there'd be a singularity at the tip.

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Well I meant just the shape in 3 dimensions assuming the universe could somehow come out of the Big Bang shaped like an actual cone (in reality it's probably spherically distributed, isotropic and all that). And regarding global topology versus shape of the metric, I think you're confirming what I asked. The metric is sort of describing the "worldline" of the entire universe then? –  Ocsis2 Apr 10 '12 at 19:48
So there are 3 shapes to consider?: Shape of the universe in 3 dimensions = Spherical (Probably) | Shape of the topology (current global intrinsic curvature at one instant of time) = flat | Shape of the FRW/RW metric/RW spacetime (somewhat like a worldline), some kind of weird cone representing the evolution of the universe's spatial dimensions (and intrinsic curvature) with time –  Ocsis2 Apr 10 '12 at 19:48
Forgot to add, the "flat" shape that the FRW metric points to is different from the idea of global flat intrinsic curvature at one instant of time, correct? Because the FRW metric represents the evolution of the universe over time so a flat shape there actually means a non-ending expansion (which when depicted visually comes out like the weird cone I was talking about). –  Ocsis2 Apr 10 '12 at 19:56
I'm guessing you're a bit mixed up about how the universe evolved after the Big Bang. The universe didn't start from a point and expand like a cone. The universe has (probably) always been infinite. All that happens as you work backwards towards the big bang is that the scale factor decreases, but the universe remains infinite. –  John Rennie Apr 11 '12 at 10:33
Infinite how? In volume and size? Also, are you distinguishing between the universe and spacetime or are you saying spacetime has always been infinite? –  Ocsis2 Apr 11 '12 at 15:19
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