Could this suggest that there is a wavelength smaller than Planck's? Suppose the earth receives a photon with a wavelength $\gamma_1$. Since spacetime is expanding, we know that this photon had an original wavelength $\gamma_2$, such that $\gamma_2\lt\gamma_1$. This is phenomenon is known as redshift. Nothing special. 
Now, here's the thing, what if the earth receives from far away a photon whose wavelength is equal to Planck's length $\ell_p$. This means that the photon before traveling all this distance a wavelength smaller than $\ell_p$. But this is impossible, since $\ell_p$ is the smallest possible wavelength. So would this suggest that there is a wavelength smaller than Planck's?
 A: What you are describing is known as the Trans-Planckian problem. Known examples of this are in Hawking radiation and inflationary cosmology - in both cases we end up having to consider wavelengths shorter than the Planck length.
As far as I know this is still an open problem. If you Google Trans-Planckian you'll find no end of articles discussing the problem and various solutions to it, but no consensus on possible solutions has been reached. The problem is that we have no theory of quantum gravity that can describe what happens in these regimes.
So I'm afraid the answer to your question is that no-one knows the answer to your question!
A: Yes, according to the standard theories. Anyway, neglecting the technical problems of detecting this photon, at Planck scale (and beyond!) you are in the regime of Quantum Gravity, so there isn't a final answer at the moment.
A: You have to distinguish mathematical models from real world. Physics tries to model the real world with the best accuracy possible, and having a theory that works at some scale, doesn't mean that you can just infinitely extrapolate it to infinite extent.
A famous and well-known example of this predicament is the Newton laws, where people thought that because they work at some framework, they're universal, and then we discovered we're wrong in two things! First, relativity, where we saw that objects approaching the speed of light don't follow simple Newtonian mechanics; and second, the quantum world, where Newton believed in determinism, and we found then through Quantum Mechanics that nothing is deterministic microscopically and we have branching ratios for almost every event.
The Planck scale physics is suggested to be very different from the physics we know. At that scale, all physical forces are equivalent and have the same proportionality constant.
That said, the decent answer to say is: I DON'T KNOW.
It's very wrong to extrapolate infinitely, and we have to draw the line between science and philosophy, and the way I see it, the issue you posed is philosophical and is a nice question, but as long as it's not in the testable range, we shall not pose answers we can't verify.
