If I build roads in a friendly civilization, are they just free, or does that civilization pay for them, and if they do pay for them is it possible to encumber their economy by filling their nation with roads?
There is another similar but broader question here. The answer left unresolved your particular question - what happens to roads you build in other players borders.
So you can theoretically wage economic warfare by building roads in other players land.
Also another nugget - roads that used to be in a civilization's borders (aka when a city is razed) are free - nobody pays for them.
You pay for roads outside your territory - 6 roads here
Also, some quick digging around in the SDK source code. Since my digging was somewhat cursory, there might be some more edge cases around. The relevant code files and functions in the SDK are:
You pay for all of your roads, whether they be in your territory, neutral territory, or allied territory. The upkeep is also paid by you.