If you actually try to demolish a building, it will prompt you, asking "Do you really want to demolish <building type>?" You can then accept or decline. As far as I know this is the only way to tell what an individual building is.
I think it would be helpful if you could hover over the buildings at any time and have a pop up in the UI telling you what they are.
I definitely agree with this. As a city builder, I'd like to know (not that it matters except in my own mind) that I don't want to place the gallows right next to the merchant or inn, might depress commerce. Also, when I conquer a city I'd like to have an easy way of discovering what buildings are already in that city without having to go to the city details view.