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: 
This goes on order from the more general to the more specific:

1. If A and B are two states, then you can form any linear combination to get superposed state, let's call it C. Of course, any state C can be expressed as superposition of two other states A and B, so calling a state "superposed" is not all that meaningful, unless there is something distinguishing the A,B states from all their superpositions.
2. You can only discuss entanglement if you first distinguish two (or more) sub-systems of the full system, for example if the distinguished sub-systems do not interact with each other, or if they are spatially separated. In this case states of system (1) can be entangled with states of system (2), if the total state cannot be written as a product $A=A\_1 \times A\_2$ where $A\_1$ and $A\_2$ are states of the separate subsystems. A more formal way to quantify that is the "entanglement entropy", which tells you roughly how much information you are missing if you have access to one sub-system only.
3. The cat state is a particular example of an entangled state. Specifically in systems where two (or more) sub-systems consist of exactly one (qu)bit of information, which is labeled as 0 or 1, this is the state |00...0> +|11...1>. I am also told in comments there are generalizations beyond this system of n qubits, and more physical ways to characterize the state.

So, if you decide that your product states as above are distinguished over any linear combination thereof, there is a meaningful comparison between what you call "superposed" state (which I have to say is not a standard terminology) and the entangled states. An entangled state is one that cannot be expressed as product state, so it is necessarily a superposition of such states, say $A\_1 \times A\_2$ and $B\_1 \times B\_2$. The converse is not true: a superposition of product states may well be a product state itself.
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.