When a really faraway city state asks me to build a road to join our two cities, is this a bug or is there a way to do it?
In general it isn't worth it, but if you build a city with a harbor nearby, it can be done without a ridiculous expense. Since a harbor extends your trade network the same as a road would, you would then just build a road from the harbor city to the city state.
As long as you're not hostile, I believe you can make a road through that civilization. Just get an open border treaty. I'm not sure who pays for the roads after they're built though. That I'm still trying to figure out.