We already have a sort of demonstration of this when we suddenly opened up travel from being (generally) foot-bound to the motor vehicle. I.e. suddenly massively opened up the space available for travel in a given time.
Upon invention, congestion will become a thing of the past, briefly. But then, as always, traffic will increase to fill the available space (yes, there might be a lot of it going up but there is still going to be some sort of flight ceiling). Even if "roadspace" is no longer the general constraint, you've still got to park those cars somewhere. Everyone has to start and end a journey, so those become the constraints (were they not already). For a sense of the effect, about 50% of Los Angeles land take is transport infrastructure/parking lots and that's not "solved parking". Where are all these new parking lots going? Even if you build up on parking, you've still got to get into and out of it; think pedestrians near a busy subway or rail station. 461
The basic principles of crowding behaviour apply as much in 3D as they do in 2D.
From the urban design perspective, I don't really think anything changes. We spent half of the 20th Century ploughing highways through city centres and building huge amounts of vehicle infrastructure, only to realise that it's not workable/desirable in the long term and move back to people-scale infrastructure and urban densification with vehicle access restrictions and parking provision restrictions, and a focus on public transport provision.
So, at worst we go through another cycle of that process again. More likely is it becomes an excuse to stop maintaining existing road surfaces which now become pedestrian and cycle routes (and maybe public transport). Vehicle traffic is moved off the surface (up). Access and movement within buildings is still on foot so we aren't abandoning elevators, as a result the primary constraint on general building height still remains. Furthermore, the other main constraint on building height is the value of the space within; nothing about this changes that (if anything the potential for sprawl devalues it).
Assuming the flying cars aren't noisy, I'd say, not a lot changes.
Re jetpacks, I think all of the above still applies but look somewhere like Amsterdam (i.e. for what happens with mass-cycling) for the urban effect (i.e. none) compared to somewhere like L.A. for vehicles. 461