In general, I would advise not to assume that centralized systems will be prominent in the future.
The trend toward centralization in the late 1800s through the mid-1900s was largely a result of the network effect in analog communication technology giving an advantages to vertical hierarchal organization. It pretty much reached its limits by the 1970s at the latest and died utterly with the digital age. It keeps on in government largely by inertia and governments ability to force people to support it no matter how dysfunctional it becomes.
All the action in systems theory today is in decentralized systems like swarms. I wouldn't look to centralized systems of the past century as models for the future. That would be like someone in 1775 assuming that the America of 2014 would be some form of Monarchy because almost all government up to then had been monarchies.
A strong government would naturally be required, as the cost, in
resources, time, and intelligence, to maintain such an enormous
construction would be astronomical.
Even if you look at modern cities with notionally strong centralized governments, like modern day New York, you will see that the vast majority of operational decisions remain private e.g. who buys and sells what property, who pays for most construction, how the businesses who directly or indirectly pay all the taxes operate. Government remains a relatively thin gloss on top of the private sector. The private sector exist a priori to and without government but government cannot exist before and without the private sector.
A strong centralized government would only be absolutely necessary if the city planet had no actual economic rationale and the government had to forcefully direct the allocation of resources to and within the city to prevent the spontaneous flow of resource away from the planet city.
This is even more paramount if the city extends greatly both above the
natural surface and into the ground, as they would be expected to do
so once all horizontal space is occupied.
I don't see why. Skyscrapers are and have historically been completely private constructions. Even within single buildings, you can have internal private property. This is the concept behind condos and coops.
Could the city-planet ever be privately owned, or would it have to be
controlled by a centralized body?
If the entire planet was owned by a single entity, it would function internally as if run by a non-representative government. Corporations operate internally much like governments (more accurately modern government administration is are modeled on corporate operations) which is why corporation often stop functioning and go out of business.
A more interesting question is if such a city could operate primarily with billions of private owners using just decentralized, voluntary, private decision making and exchanges. Clearly it could.
As long as the planet city had a functional/economic reason to exist, it could operate by decentralized spontaneous cooperation. The great cities of the English speaking world e.g. London and New York, grew to the largest cities of their day in a decentralized manner and only became centralized as they declined in relative dynamism.
Just because people are crowded together and very interdependent doesn't mean they can't exist without Hobb's Leviathan threatening them constantly. Property systems work very well to allocation resources on a voluntary basis.
Ownership means the ability to make decisions about the allocation of a resource. Property system manage the allocation and transfer of that decision making authority. Property system can allocate any resource, not just land area.
For example, in a giant planet city arcology, a valuable "resource" would be load bearing structures. Anybody wanting to build a new structure would have to build over an existing structure. To build over, you would need to put load on an existing structure. The ability of the existing structure to handle the load or be adapted to do so would be very valuable and so could be made a property. People could sell the load bearing capacity of their property just like we sell off water or mineral rights while we continue to live and farm on the land surface itself.
Water, air, power can all be managed by property systems without centralized coercive government.
It's even quite possible that a centralized government can't scale beyond managing systems of a certain size and complexity. Centralized government by definition means fewer decision nodes which means fewer people making more of the decisions. Since individuals can only process so much information per unit of time, as the system managed gets larger, the decision making process gets slower and slower. In a vast planet city, the decision making process might get so slow as to cause the system to be effectively paralyzed. (Corporations stared decentralizing in the 1970s and many argue that large modern cities have already reached the point were they must decentralize of implode.)
By contrast, decentralized system have a huge number of nodes and make decisions in parallel. They can scale massively and get faster the larger they become. Biological systems work this way and they are far more complex than a city.
Could crime even exist if such an extremely strong local government is required?
Strong centralized governments tend to foster crime, not suppress it. When governments have no competition, they have little incentive to be either efficient or effective. Centralized governments are more prone to corruption both from within and from external criminals. Centralized government tend to evolve to see the world as divided between not the lawful and unlawful but those within the government and those outside the government. Those inside the government don't care if those outside get mugged.
The Soviet Union had shockingly high levels of even common street crime dating back to time of Lenin and they never managed to control it even though they were a police state. Their police states focused on political threats, usually from their fellow ideologues, and ignored prosaic crime. Likewise, the Russian mob didn't just spring into existence when Communism fell it was around for a long, long time. It survived by fostering corruption within the Communist regime itself. Similar patterns existed in all totalitarian states.
You can see the same effect in the complex of big city political machines, unions and mobs that highjacked many US cities in the 1920-1970s. Organized crime was massive and eventually, unorganized crime grew out of control as well even as the local governments got proportionally larger but more effective at the same time.
How would terrorism be prevented from taking out sections of possibly
kilometers-high load-bearing populated structures?
The bigger the system, the more robust and hard to disrupt it will be. It's been said that you couldn't might not be able to destroy Hoover Dam even with a nuke and couldn't even dent it with any plausible amount of conventional explosives. The support structure for a kilometers high structure would be similarly massive and hard to destroy.
A more likely target would be the informational systems that would control power, water and air. A planet city would have no natural buffer left and would function more like a spaceship. The easiest way to kill a lot of people would be turn off their ventilation.
Again, a decentralized system would be more robust. In the early days of electricity before it was socialized, there were often lots of redundant cables strung all over because many different companies each supplied power on their own cables. Sounds chaotic and wasteful and was to some degree but on the other hand, such a system made a city and region wide blackout impossible while they occur fairly often with the "efficient" centralized system.
In a planet city where interruptions in power and air could be lethal in mere hours, it would be suicide to have a single, centrally controlled system. One clever terrorist could bring down the entire system for billions. A tangled chaotic system of private suppliers of power and ventilation would be far safer.
As much as we twitch about our high technology systems, they are arguable more robust than their precursor technologies. For example, it inconceivable that a major modern city would simply burn to the ground but massive fire routinely destroyed major cities up until the late 1800s when concrete and steel replaced wood. The great Chicago fire is just the best know of several major fire that destroyed big swaths of several major US cities. Fears that enemy agents and anarchist would burn down entire cities where quite common and it was in fact tried. Had they known more about how to set fires, they might have succeeded.
Imagine people a 150 years ago contemplating a modern skyscraper. They would assume that assume it that, even if it stood, it would likely soon burn to ground. After all, it would be filled with wood, open flames would provide heat and light, fire suppression systems were a bucket of water and fire resistant materials except asbestos unknown.
We're probably in the same situation when we try to imagine a future system like a planet city.