Yup. Have fun wearing your completely soaked overcoat in a thunderstorm. If you dress professionally, you're gonna need a classic raincoat as well as a classic overcoat. They have different functions, and neither is particularly well suited to the role of the other.
I vary. Sometimes I button mine and leave the belt hanging, sometimes I half button and tie the belt in front (a la Humphry Bogart), sometimes I don't button at all and just cinch it with the belt, in which case I use the buckle. It all depends on the temperature and my mood.
The fun part of a trench coat is that you have options.