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.
Simple answer: No, not directly.
I played around with my save game, and the difference appears to be that as a result of going into "We love the king" mode, my gold from cities went down dramatically. I would guess that this is because the civilians were assigned to food instead of money to take advantage of the growth benefit. I guess this means you have to keep a close eye on how your civilians are assigned, especially later in the game when maintenance is high and money is a lot more important than a little population growth.
Best Answer
If you kill quite a number of units of a city state and/or keep trespassing on their territory enough to get your score very very low over an extended period of time, they eventually will be convinced that you are so evil it's not worth considering peaceful coexistence, so they will be in perpetual war with you. You will not be able to buy their friendship and relations will not heal by themselves over time.
Actually that doesn't mean too much, though. If you go near them, they'll attack you, and if you leave cities undefended near them, they'll capture those. Apart from that, they won't hunt you across the map and they won't make others hate you as well. If you just leave them alone now and make their area taboo to your units you can still get along with the other city states just fine.
You might choose to have one perpetual warring city-state next to your army outfitter because city-state units train your troops longer than barbarians do.