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: 
Any discussion of entanglement implies you are thinking of (at least) two separate systems. Superpositions on the other hand, can also apply to situations where you are only looking at a single system. Keeping this in mind:

> 
> "Can a state be entangled without also
>  being a superposition? (Please give an
>  example)"
> 
> 
> 

No. An entangled state is by definition a state which cannot be written as a separable product in any basis. Whatever basis you choose, the state will be a superposition of products of the separate systems.

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

Of course. A trivial example would be a single isolated system for which you can usually find some basis in which the state is in a superposition. And single isolated systems cannot be entangled with anything else because.. well.. they are single isolated systems - and like I said before, entanglement applies to a situation where you have divided your world into separate systems that can interact with each other.

> 
> "And what about a cat state?"
> 
> 
> 

A Schrodinger Cat state is usually used to refer to a *macroscopic* system which is in a superposition of two or more states (in the eigenstate basis of some physical observable like energy or position). Real cats in geiger-counter-poison-rigged boxes probably won't be in such superpositions due to [decoherence](http://en.wikipedia.org/wiki/Quantum_decoherence) but there are some surprisingly large objects like buckyballs and macromolecules which have been shown to exist in superpositions.

> 
> "Are these spin pairs also in a
>  superposition?"
> 
> 
> 

Yes, in all bases, because they are entangled. Allegedly :)

> 
> "Does this count as a cat state?"
> 
> 
> 

I'm not sure because I haven't read the whole paper and also because the definition of a cat state depends on what you consider macroscopic.
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.