To directly answer the question, road maintenance is listed as "Tile Improvement Maintenance", when you mouse over your treasury/income in the top left. You can see a circled example in the screenshot below. As for the details:
Roads cost to maintain both inside and outside your borders
Because my conclusions are contrary to previous info we had about road maintenance in neutral territory... Pics Or It Didn't Happen!
Exibit 1: Prince Difficulty, some roads in, some out.
As you can see, I have 4 roads in my territory, and one outside, costing me a total of 5 maintenance. I have no other roads anywhere else. I wish I had taken a screenshot of the previous turn. The next time I find myself in this situation, I'll take a pair of screenshots and replace this single picture. Either way, remember you can always try it yourself to be sure. YaY Science!
Exibit 2: Settler Difficulty, all roads out.
Thanks to @DMA57361's questions, I checked a different difficulty and a different number of roads outside the borders, namely all of them. On Settler difficulty, building roads only outside of my borders, I still ended up with maintenance for the roads, so clearly they do cost.
Edited to account for difficulty, now that I've completed more extensive testing.
Roads cost the same to maintain in your own territory, and in neutral territory. How much varies by difficulty:
Road Maintenance /Road/Turn
Difficulty | Your Land | Neutral
-----------|-----------|--------
Settler | 0.34g | 0.34g
Chieftain | 0.50g | 0.50g
Warlord | 0.75g | 0.75g
Prince | 1.00g | 1.00g
5-7* | 1.00g | 1.00g
Deity | 1.00g | 1.00g
Road maintenance is only ever a whole number. In case of decimal remainder, always round down. For example, on Settler, 2 roads cost 0.68, which clearly got rounded down to 0.
To find each of these, I built roads either only inside, or only outside my borders at a time. This confirmed the same results for inside and out, but got me only close numbers. Then @Oak used his powers of XML scrying, and found the exact numbers for us.
Since so many difficulties are 1g, (and the ones that are not are the easier difficulties anyway,) and for our general happiness, I hereby declare "1g" an acceptable approximation for all further calculations relating to road maintenance. Hurray!
When next I can test, I'll re-test road's in another nation's borders, to clean up the next section! Same Bat-time. Same Bat-channel.
Roads DO NOT cost to maintain inside another nation's borders
Edit: This section is suspect until I complete further testing, since one road did not cause me maintenance. My mistake for not setting up a proper control. This still may be completely true, but I need to test on different difficulties and compare with similar numbers of roads in my borders and in neutral territory.
In the game these two pictures originated from, I had no roads, and built a road inside each another civilization's borders, and a city-state's borders. I can't verify that they were costing any gold to the land's owner (Rome's income was fluctuating quite a bit), but I can say for sure they were costing me nothing. This is likely what bwarner saw that caused the error.
There may be other effects that change which roads cost to maintain, but these seem to be the major ones. Anyone found anything not explained by these.
PS As with any screenshot heavy post, I do my best to crop/scale the images to reasonable sizes without losing the important parts, such as the circled text. They still end up crowding the page, for which I'm sorry.
Best Answer
Just to set the stage, "influence" is the way a city-state feels towards you, and you gain influence through gifts and by doing quests for the city states. There isn't a single source for "reputation" or any easy way of tracking it globally. You do have certain modifiers which each civ which could be considered a reputation of sorts.
Declaring war on city states does impact this per-civ reputation, and in some cases it can permanently change your influence with the city states. If you go on a city state conquering rampage, city states will start to band together and declare war to try and stop you. Further, your reputation with the other civs will take a major hit in many cases.
Declaring war on civilizations can effect your influence - city states allied with the civ you declare war on will also declare war on you and their influence will immediately drop all the way to the minimum number it can be. City states that aren't allied to anyone you're at war with don't tend to care, though.
Essentially, if you go around declaring war and conquering civs, get used to being denounced by other civs. If you eat city states for breakfast, get ready for city states to permanently hate you, and you'll be denounced yet again.
There are some ways to get your war on without taking major hits to diplomacy - this question and its answers go into more detail on that subject.