Is "map" also an appropriate translation of план in this instance, for example referring to the city's road map, or would that be limited to карта города? To me as a native English speaker "the plan of the city" refers not just to the road layout, but where buildings are located, parks, utilities, etc. It also has the sense of being what is to come, rather than currently present. The average person would never be involved in the plan of the city, whether in the abstract sense of planning, or possessing the plan in document form.
Yes. In Russian, "план города" can mean both, but unless you are an architect or a city planner/engineer, you would probably be talking about a city map. And for that "карта города" can be used equally well.
There is a very slight difference between "план города" and "карта города" though: a plan can be more schematic, while you would expect more faithful representation form a map. So I would probably use "план города" for something I find in a guide book and "карта города" for a good folding map. These are real subtleties though, and I am not even sure my preferences coincide with other people's usage.
If you are asking whether the origin of this word is that same as that of English "chart", "card" and Latin "carta" - I am sure it is, but it most likely made it to Russian via German "Karte", which has a number of meanings coinciding with the Russian use (e.g., a playing card).