No. Resources yes, like you mentioned but regular improvements not really. However note that it's a massive PITA when you're getting pillaged and you can't actually GET units to the tiles fast enough because you didn't build enough roads (er... not that that's happened to me or anything)
You need roads to connect a resources (copper, corn , gold etc) to use that resources. Example, you can have you citizans work the farm for 5 or 6 food, but do not get the +1 health unless it is connected via a road or river (you need sailing to use rivers as roads).
A non-resource mine, farm, or cottage does not need a road to be useful except to speed up travel. The one exception is building a railroad on a mine, lumbermill, or quary gives you an extra +1 hammer.
Providing it is within your fat cross, you can work any farm, mine, camp etc and obtain the food, hammer and commerce benefits without having a road connecting the resource square to your city. However, you will not get be able to use the resource (for example to build axemen using copper) unless you have a connection by road or river. Neither will you get any happiness benefits (from gems for example) unless connected.
As stated above, roads have 3 uses
1. move faster
2. connect resources
3. trade routes
The first 2 are well known. Concerning the 3rd, it might be worth having a look at your trade routes in the city screen. When a city is connected to another one, it can establish one or more trade routes, which yield extra commerce. Foreign trade routes are worth more than domestic trade routes, so yes, it's a good idea to have a road connecting your empire to foreign nations. Note that rivers also do the job. Also note that you need an open borders agreement for these trade routes to work. That's why it's a good idea to sign open borders agreements.
Not quite sure about tha, to be frank. I know that's what the Civilopedia says, but i'm pretty sure i get the "congratulations, you've connected your first city to your capital" whenever i build my 2nd city on the same river than the capital.
One thing though: it is possible that the "road" formed by the river needs to be entirely in my culture for this to work. So maybe Sailing allows rivers to function as roads regardless of the culture.
Trade access to another civ can spread religion. Spread is slow without open borders, but no trade access stops the spread of religions completely. If a religion spreads to your civ and you don't have trade access to the source civ, it had to have jumped through another civ that does have trade access with you. (You don't have to have access to them... they might have sailing and coastal trade access with your neighbor... and by extension with you, for example. You don't have to have either.)
You can't open foreign trade routes to other civs without roads or water access (roads or rivers over land, sailing to activate river and coastal trade, astronomy for ocean trade). That first trade route you start with is +1 to your own cities early game, but with a foreign civ it can net +3 or +4 commerce each (and up to +7 or +8 later in the game, for EACH trade route in EACH city). Trade routes get more numerous as you research specific commerce techs.
That's all I can remember right now.
edit - yeah, I once built a second city further down the coast, and got the "congratulations, you've connected your first city to your capital" message the exact same turn the cultural borders closed the gap in the coastal tiles, even though I didn't have sailing yet.
edit - yeah, I built a second city farther down the coast and get the "congratulations, you've connected your first city to your capital" message the exact same turn the cultural borders close the gap in the coastal tiles, even though I don't have sailing yet.
i had that along a river when my second city got the culture pop. bizarre thing was that they didn't share all resources, just the one between the two cities until i got a road in there. i wish i had a save to figure out why the heck that was. maybe i had to get the road to the resource south of 2nd city, which was connected to 2nd city by a river only at that point?