I always assumed that a court halved the maintenance number in the upper left corner in the city view. Is that right? And if so, then a court in your capital is useless, correct?

Your capital has maintenance costs (the amount of maintenance in a city depends on 1. Distance from a 'Palace' and 2. Total number of cities)

So once you have enough cities your capital will have maintenance costs

Yes, your capital will get maintenance costs, although it will rarely be more than 2 or 3 gold. Still, you can probably build a courthouse very quickly, they're very cheap in the late game, so it may be worth it.

I always build a courthouse in my capital when I have pretty much run out of other stuff to build. Its definitely not a priority tho. As oagersnap said, the maintenance in the capital usually doesnt run very high but saving an extra gold or 2 cant hurt you any, but it really isnt a priority. :)