Most true suburbs aren't within the big city limits, so I'm not sure your point is well-founded. For example, in the DC area, the suburbs aren't even in the same state as the city and yet the suburbs seem to be thriving.
Those suburbs outside the city limits still need money. They get it from state and federal funds which were mostly collected from people in the city limits. For example the Highway Trust Fund as one of many examples. If you check the per capita spending, it's higher for suburbanites than urbanites but the urbanites are putting in more money.
Absolutely not in the case of the DC metro area, on both the MD and VA sides. Those counties subsidize DC proper in various ways, along with the less populated portions of their own states, and because the median household income is so high, they also pay a disproportionate amount of federal income taxes when compared to DC residents.
I also have no idea why you think city dwellers are the primary contributors per capita to the Highway Trust Fund which is funded via a tax on fuel (i.e. miles driven).
DC is an exception, most American cities have large swaths of suburbia within the city limits and even larger ones within the same state.