Question: 1) Can a state be entangled without also being a superposition? (Please give an example.) 

2) Can a state be a superposition without being entangled? (Again, an example please.) 

3) And what about a cat state? 

I am royally confused, and a little bit of googling didn't help.  

This question is partly motivated by the recent news announcement (Nature, NYTimes) that ten billion entangled pairs have been created. 

4) Are these spin pairs also in a superposition? 

5) Does this count as a cat state?

Answer: 
The difference between a superposition and entanglement is the following. We consider a two slit experiment where a photon wave function interacts with a screen. The wave vector is of the form
$$
|\psi\rangle~=~e^{ikx}|1\rangle~+~e^{ik’x}|2\rangle
$$
as a superposition of states for the slits labeled $1$ and $2$. The normalization is assumed. The state vector is normalized as
$$
\langle\psi|\psi\rangle~=~1~=~\langle 1|1\rangle~+~\langle 2|2\rangle~+~ e^{i(k’~+~k)x}\langle 1|2\rangle~+~ e^{-i(k’~+~k)x}\langle 2|1\rangle
$$
The overlaps $\langle 1|2\rangle$ and $\langle 2|1\rangle$ are multiplied by the oscillatory terms which are the interference probabilities one measures on the photoplate.

We now consider the classic situation where one tries to measure which slit the photon traverses. We have a device with detects the photon at one of the slit openings. We consider another superposed quantum state. This is a spin space that is 
$$
|\phi\rangle~=~\frac{1}{\sqrt {2}}(|+\rangle~+~|-\rangle).
$$
This photon quantum state becomes entangled with this spin state. So we have
$$
|\psi,\phi\rangle~=~e^{ikx}|1\rangle|+\rangle~+~e^{ik’x}|2\rangle|-\rangle
$$
which means if the photon passes through slit number 1 the spin is + and if it passes through slit 2 the spin is in the – state. Now consider the norm of this state vector
$$
\langle|\psi,\phi|\psi,\phi\rangle~=~\langle 1|1\rangle\langle +|+\rangle~+~\langle 2|2\rangle\langle-|-\rangle~+~ e^{i(k’~+~k)x}\langle 1|2\rangle\langle +|-\rangle~+~ e^{-i(k’~+~k)x}\langle 2|1\rangle\langle-|+\rangle.
$$
The spin states $|+\rangle$ and $|-\rangle$ are orthogonal and thus $\langle +|-\rangle $ and $\langle-|+\rangle$ are zero. This means the overlap or interference terms are removed. In effect the superposition has been replaced by an entanglement.

This analysis does not tell us which state is actually measured, but it does tell us how the interference term is lost due to the entanglement of the system we measure with an instrument quantum state. So one does not need to invoke an outright collapse to illustrate how a superposition is lost. 

The cat state is a form of the GHZ state where you have
$$
|\psi\rangle~=~C(|0\rangle^{\otimes n}~+~|1\rangle^{\otimes n})
$$
which are n-partite entanglements. These types of states can violate Bell inequalities with one state, which illustrates quantum mechanics is not a purely statistical theory. The statistics are derived from QM.
Superposition involves linking multiple particles via a single dimensional thread of wave energy. In contrast, entanglement involves the separation of space-time; i.e. a single point in space split into more than one location.