Is this a "book opening"? If so, what's its name?

The position of C/D pawns, the Knights and the Bishops are positions I have found myself in on several occasions and I have met this position with mixed success.

I'm just wondering if it is one of those "book openings" and if so, what it's name.

I'm also wondering what's wrong with it and what's right with it.

If you play c4 it will most likely become a Reti/English and if you play e4 is will be a King Indian attack i think.

ECO CODE is A07: Reti Opening - NY and Capablanca Systems

The ECO Code is more important than the name, many players will call this a Kings Indian Attack based on White's set-up. But the ECO Code is always the same no matter what name you attach to an opening

Kings indian attack.

you want to know the name of blacks line?

Yes, Rumpel. Black's line.

White really does not want to delay d3 after black plays e5.

Openings are not really named from one side's play. Although generally if you are in that set-up than you most likely in a reversed pirc or Reti opening.

It's a king's indian attack.  Next white should play b3 or e4.  b3 puts the dark square bishop on a nice diagonal attacking the king.