Does String theory say that spacetime is not fundamental but should be considered an emergent phenomenon?
If so, can quantum mechanics describe the universe at high energies where there is no spacetime?
Schrodinger equation assumes the presence of time dimension. So how can such equations be valid at extremely high energy?
And is it meaningful to do physics if there is no time dimension?
Would the concepts of Hilbert space, wavefunction collapse etc be useful in that case?
The question of how to do physics when the notion of space and time breaks down was a major worry of Heisenberg in the 1940s. He thought space and time already breaks down at nuclear scales, because the proton isn't pointlike. His resolution to the problem of doing spaceless timeless physics was to use Wheeler's S-matrix as the fundamental variable for physics, to do a pure S-matrix theory.
The justification for S-matrix theory is that while we don't know how space and time work in the microscopic theory, we know that the macroscopic theory has certain symmetries at large macroscopic scales, away from the problematic regime. We have translation invariance and rotational/Lorentz invariance. This allows you to define asymptotic particle states on macro-scales, which are defined as eigenstates of the Hamiltonian (limiting eigenstates, in the sense of plane-waves), and define their scattering. The point is that the S-matrix in-states are always well defined as boosted stable particles in the infinite past, regardless of how horribly spacetime break down during the intermediate stages, and the out-states are similarl well defined. So there is no problem of principle is giving the S-matrix at all energies and all momenta for all collections of particles, without ever directly mentioning space and time inbetween.
The basic idea for constructing theories without space and time, and the answer to your third and fourth question, is to shift from a space-time theory to an S-matrix theory. Chew advocated this point of view for the strong interaction, and in the late 1950s and early 1960s, this was the dominant approach to hadronic physics. Stanley Mandelstam defined extra relations on S-matrix theories which allowed you to make restrictive conditions on the S-matrix from the spectrum of the theory, the allowed particles.
This was the way string theory was discovered by Venziano and others in the 1968-1974 era. This is not emphasized in most modern treatments, because most of the development of string theory from this starting point was devoted to working backwards--- now that you have an S-matrix theory, you try to reconstruct the space and time again to as great an extent as possible.
In string theory, as it was originally formulated, you didn't have space and time variables (on our space and time), you only had an S-matrix for the collision of various particles, which could be reconstructed order by order in a string expansion. The string expansion did have an infinite tower of resonances, and this allowed Veneziano and others (including Fubini, Ramond, Nambu, Susskind, and Nielsen) to identify an internal space, which eventually turned into the string worldsheet. Mandelstam and collaborators showed how to formulate this as a field theory in light-front coordinates, which reconstructed all but two of the space-time coordinates that the string is moving in explicitly, so that the theory was almost formulated in space and time completely.
The reconstruction of the string worldsheet motion in space-time was studied by many authors, culminating in Polyakov's formulation of the perturbation theory of strings as a path-integral over string trajectories in a normal space-time with background fields. Friedan showed that the asymptotic behavior of these background fields is what you would expect from classical massless theories evolving in space-time, with supersymmetry. The result of this work was that people tended to renounce the idea of S-matrix theory, saying that strings are just space-time moving linearly extended objects.
So string theory in the 1980s was pretty conservative with regard to how space-time is to be considered, since it seemed that almost all the space-time is still there. This wasn't completely true, though. None of the approaches actually reconstructed the full space-time at all distance scales, they simply used space-time as intermediate variables in the calculation, where you sum over world-sheets. The world-sheet sum is not so sensitive to small-scale things in the spacetime, so that people were able to do discrete large matrix reductions of certain low-dimensional string theory, where the full string theory emerged from string-bits.
It is also true that the reconstruction of space-time constantly hit a brick wall. For example, one approach that tried to produce a full space-time description is string field theory, and in the end, it is defined on a loop space, and the way in which you check that it works is by reproducing the string scattering expansion order by order, you don't have a non-perturbative way to calculate string field theory to all orders (at least not in a way that is just as murky about how much space-time is left as in any other approach). Further, the string fields don't obey microcausality (because they are on loops), and the theory doesn't clearly make sense non-perturbatively.
So the S-matrix character of the theory never really went away, and in the 1990s, it became clear why. The string theory has a property of holographic duality, which means that the spacetime near a black hole is reconstructed from the boundary data. The precise formulation of holography in Matrix theory and AdS/CFT showed that one can give many alternate formulations of string theory, which all have the property that the spacetime in the bulk is reconstructed from boundary space-time (in the case of matrix theory, you reconstruct it by a large N limit of a one-dimensional quantum mechanical system).
The flat-space limit of AdS/CFT boundary theory is the S-matrix theory of a flat space theory, so the result was the same--- the "boundary" theory for flat space becomes normal flat space in and out states, which define the Hilbert space, while in AdS space, these in and out states are sufficiently rich (because of the hyperbolic braching nature of AdS) that you can define a full field theory worth of states on the boundary, and the S-matrix theory turns into a unitary quantum field theory of special conformal type.
So the end result is that string theory always reconstructs space time from a boundary, whether it is S-matrix or CFT boundary depends on the asymptotic boundary conditions. The question of deSitter boundary conditions is most interesting, both because it is unsolved, and because the universe we live in was deSitter in the past, during inflation, and looks like it will be deSitter in far future. The question of what the appropriate boundary theory for deSitter spaces requires a new idea, and there are many proposals, although none is fully persuasive today.
The answer to your 4th and 5th question is not so simple, because S-matrix theory is so difficult to reconcile with intuition. Within an S-matrix theory, there is still a Hilbert space, defined by asymptotic in and out states, and in principle you are supposed to imagine every state of the world as a superposition of incoming particles which would produce this state "now" (even though the "now" is not completely well defined, the superposition of the incoming states is well defined). Then a measurement is a projection of the "in" state in response to an observation, which can be viewed as a selection of which branch of ridiculously complicated macroscopic "in" state superposition we have found ourselves to be in.
The same is true in AdS/CFT. Any state of the interior of an AdS quantum gravity theory should be thought of as a state of the boundary field theory, and then the measurement phenomenon is just the same as in measurements in ordinary quantum field theory, or in ordinary quantum mechanics.
The only difference with ordinary quantum mechanics is that the classical limit is removed from intuition even more by an extra layer of abstraction--- shoving every state to an asymptotic "in" and "out", or to the boundary CFT. But if you are comfortable with the process of measurement in ordinary time-coordinate quantum mechanics, you shouldn't be less comfortable with it in S-matrix theory. The only question is how comfortable you should be with it in ordinary quantum mechanics.
Not in the usual description of string theory. Spacetime enters the theory by stating that a string has coordinates in spacetime, so in this sense it is a fundamental property of the theory.
Some string theories can be described in an alternative way, by a field theory (not a string theory) in which there is one fewer spacetime dimension. This is called holography. In this alternative description, you can say that this 'extra' dimension is emergent because it is not part of the spacetime of the holographic field theory.