Why does the moon orbit the sun?

I don't know what 'initial momentum' is. As far as I understand, both the Earth and the Moon have angular momentum which they would need to get rid of in order to fall into the Sun.

 

I aways imagine the earth and moon as a single item orbiting the sun

 

That is indeed one way you can imagine it if you aren't worried too much about accuracy.

 

That's how we mostly think of it. However, the situation can arise where you get a so called 'horse shoe' orbit where the orbit of a small satellite follows a horse shoe shaped path which lies near to the Earth's orbit. There are a number of asteroids that actually do this. See this Wiki link. It is a moot point whether you would consider the satellite as orbiting around the Earth or the Sun. This could be another example of where insisting on classifying the orbit as one or the other is more trouble than it's worth.