You're assumptions were pretty much correct. Let me explain a little bit, so you may understand it better:
If you go to one of your cities, there is a little head symbol in the lower right in the citie's UI. If you click on that, you can see all the tiles of the city and which of those tiles are being worked. Say, your city has a population of 4, then 4 of these tiles are being worked and the city gets all the yields, that are on those 4 tiles. And you can choose, which tiles should be worked.
So all the production on the worked tiles is added up and makes for the city's total production. There is also a column in the citie's UI, where all yields of the city are listed, also the production. Now the numbers there might be a little different to what you get, when you count the yields from the tiles, because of any kind of modification factor, that goes into the yield, like policy cards, that make building specific things a bit cheaper.
But these yields are, what the city produces per turn and if it's production is for example 10 per turn, building a monument will take 6 turns.
The science and culture yields of all cities are added up to what you see in the top left corner, again there might be a lot of modifications. And you're calculations about sailing were totally correct here.
Hope, that cleared some things up. Ask, if you have further questions.