Trying to understand improvements

No. The only improvements worth building outside the city radii but within your borders are mines, farms, wineries and plantations on a special resources so that resource is added to your trade network. Cottages need to be worked by the city to be of any benefit. Mines also need to be worked to provide the chance of discovering a resource, building a railroad through a mine will also provide no bonus unless it's worked by the city. Roads can be built anywhere regardless of cultural borders.

EDIT: Forgot about pastures, you can build them outside the city radii too. In a nutshell, build the improvement that will provide the resource to you and it's health or happiness bonuses. As for workboats and resources in water tiles, I would only build boats for those resources that are inside the city radii. For resources unreachable in water tiles I think building workboats for those are not worth it.
 
You can harvest special resources outside of a city's fat cross, those are usually worth putting the mine/boat/etc. on. I normally workboat every sea resource unless I've got a ton of extras, workboats are cheap once you've got an empire going and it's nice to have them for trade or so you don't lose one if a pillager sneaks in.

Other than special resources, the only thing worth building outside of city raidii are roads, railroads, and linking farms. Roads and railroads help movement, ideally you'll have a road/rail on every space in your empire and it's pretty easy to do by automating workers once you're done with 'real' improvements. Farms can chain irrigation outside of a city radius, most of the time you don't need to do it but occasionally a couple of farms will chain irrigation to a town that really needs it.
 
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