First of all, Neutral Roads definitely need/should cost money... if you want a shiny Roman road going through neutral lands, you'd have to maintain it, i.e. have people on the pay roll to keep it in good shape. Obviously, whoever builds it will pay for it. And if it's too expensive, then you can destroy it.
Secondly, the main military benefits of road use, I think, is through rough terrain (hills, forests, etc.). I think it's pretty clear that if you build a road through a forest that your troops will move faster through it, i.e. as if it were open terrain... I think I saw that confirmed somewhere. What's not clear is whether a road on open terrain will increase normal movement. So, should you also be able to use enemy roads? I'd think so. If they don't pillage them, then the road should be there.
Thirdly, I'd assume that even if you can't use the road for movement, that if you place a unit on an enemy road connecting a trade route then you'll cut that trade route for that turn, even if you don't pillage the road.