Could she have been built in San Francisco? Yes, and that wouldn't necessarily preclude her being made operational somewhere else.
...Especially in the context of Prelude to Axanar, where putting the final touches on the Constitutions is done at a specific location as a strategic ruse. Few organizations in history would have built their war machines at the vulnerable front lines if they had a choice, certain siege engines perhaps notwithstanding. Starfleet here would probably have a choice: if Axanar suddenly became vulnerable, the construction yards or at least the half-built ships could be towed elsewhere. But Garth decides they should not, and the fate of the Klingons is sealed...
Unless you're in geosyncronous orbit, you don't ORBIT above a specific spot.
In today's terms, you can indeed orbit above a specific point, or fly a figure-eight, or a figure that spells "I don't brake for Newton". Orbiting is a term commonly used for flying an aircraft in a repeating pattern, for purposes of waiting (as few aircraft can hold still while waiting) or for maintaining steady contact with a surface feature such as communications asset or weapons target.
Would a starship construction facility wish to fly figure-eights above San Francisco? Well, it probably would. After all, transporters are line-of-sight devices with little or no ground-penetrating capacity, so surface-to-orbit logistics would be insufferably interrupted by freefall-style orbiting, or then need to be routed through a global emitter network. And the latter might not be treknologically or economically viable.
Could a dockyard fly figure-eights above San Francisco? I don't see why not. Starship impulse engines seldom fail. And when they do, starships fall from the skies, as if the engines had been the thing holding them up, and not Newtonian freefall mechanics. And if impulse engines are too unreliable to float your dock, you can go for antigravity, which never fails.
Is there a particular reason it couldn't fly in a planet's atmosphere?
With as much propulsive oomph as a starship obviously possesses, I don't see how any "reason" would have a snowball's chance on Vulcan of hindering flight. Aerodynamic shape non-optimal? Irrelevant - the engines can push anything through air at any speed up to maximum warp. Sonic effects a problem? They would be at any shape or size or indeed speed, for something that big - but shield tech might reroute air to alleviate the effect, or then the neighbors just have to grin and bear it. Friction effects threatening to melt you down? ROTFL - these ships can shrug off disruptor beams or the heat of stars.
Timo Saloniemi