I recommend not "investing" in a city-state unless you can get some influence for free to start out. Killing Barbarians or doing another quest for them will get your foot in the door, and let the $250 you have to spend periodically give you a lot more value. For instance, that $250 will be keeping you above the "Allied" line instead of just barely making you friends with the city-state.
Likewise, as mentioned above, the Patronage tree will give you significant bonuses to your influence over city-states.
It seems that the design is such that in the early game you won't be able to guarantee a city-state ally without committing a lot of cash.
To elaborate (not 100% sure on all these numbers but the overall point is still valid):
If you spend $250 to become Friends with a city-state, starting from 0 influence, you'll get 35 influence, making you barely Friends (30 influence). This means you'll be friends for 5 turns, at which point you'll need to spend another $250 to be friends again. You probably don't have another $250 that soon, though, so the influence runs all the way down close to zero before you're able to spend another $250, which again only gives you a few turns of friendship. This is obviously unsustainable.
If instead you perform a mission for the city-state and get 30 or 40 influence to start with, every time you spend $250 you'll get to use all 35 points of that influence before needing to spend another $250.
Similarly, you'd be better off saving your money to start with $500 worth of influence than buying $250 and not being able to afford more right away.
Long story short: if you're already friends, you get more value for your money. Likewise, if you're already allies, you get even MORE value for money you spend to sustain at that level.
Regarding passing through another civ or city-states territory:
Trade routes can pass through other civs as long as you have open borders agreement from them. Trade routes do not work through a civ you're in war with, even though technically your units can pass within their borders.
Regarding city-states, it works of you are either friends or allied with them.
Regarding passing through an actual city:
I believe it's precisely the same as the above - your road passing through the actual city shouldn't make a difference, except for saving you money, since cities count as road tiles.
Best Answer
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.