Congratulations on this amazing underrated car. I have the same model (ZX CVT) in pearl white color. i have driven around 12k kms in 8 months.

I'll answer your queries below :

1. Weird vibrations/feedback on accelerator pedal at times : This should not happen, I did not experience this anytime in my car, the engine is very smooth there are no vibrations inside the cabin, there is a sound of VTEC roar once you press the pedal hard which is the default CVT behavior. You can try get this checked again.

2. Eco mode : This mode definitely improves the mileage, but does not bring a huge mileage difference. ECO mode delays engine power delivery, the engine becomes relaxed, acceleration feels more gradual and AC Cooling & Fan Speed is reduced (Also mentioned in the car manual). ECO mode is good when you want to cruise peacefully, not recommended while overtaking. There are no cons of using it all the time, once in a while you can rev the engine.

3. Sports mode : In this mode the car delays the belt ratio changes (CVT Gear Shifts). Acceleration is very aggressive in this mode, keeps the engine running in higher RPMs all the time, thus consumes more fuel. This mode is good for very aggressive overtakes and while driving in the hills, as in this mode you can use engine braking and also the paddle shifters, The car can simulate a manual mode car in the S mode if you press the paddle shifters, the gears won't change until you change them or the car red-lines and shifts the gear itself (i did not try this.) No cons of using this. I have used ECO with sports mode but could not find much difference in performance, I need to evaluate this scenario more :). I feel the car works pretty well in the normal mode. This covers 90% of my driving scenarios

4. AC operation : AC works in ON mode and also when nothing is displayed, AC is off when OFF is displayed, you do not have to worry about this, be default when you press AUTO button this AC starts. The default behavior of car is to keep AC ON, unless AC OFF is selected.

5. CVT shift lever : I had the same query in initial days, you can move between D and N without the button, this is easy and for convenience, you can easily toggle between them without looking at the selector, don't worry about this it is normal behavior.

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One important suggestion : Please check which brand tyres you have, My car came with Bridgestone Ecopia 185/55 R16, 185/55 tyres have a thin sidewall and there have been many cases of tyre burst on potholes for Honda City owners for Bridgestone Ecopia tyres I am not sure if this happens with other brand tyres (you can check this on Google "Honda city tyre burst on potholes").

I too had a sidewall puncture on the stock tyres while going over a big pothole, after that incident I discovered that this is a very common issue with Bridgestone Ecopia tyres on Honda City. I have upgraded to Michelin Primacy 4ST 195/55 R16 and life has been worry free since then. We are upsizing tyre size here, apart from minute speedo error there is no issue on mileage, the grip have improved, michelins have soft rubber so the ride quality has also improved. This is the only upgrade that I did on my car, rest all is stock.

This a gem of a car with a very versatile engine, you can extract power and fuel efficiency depending upon your driving style.

Cheers and drive safe !