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Heavy emphasis on importing raw materials is often required by the city-planet in such worlds. Could enough material ever be brought fast enough to support the obviously enormous human population? On the flipside, could enough waste ever be removed fast enough? Where would it go?
Once the city-planet is completely built up, it would be very difficult to mine ore with which to make repairs to the extant buildings; presumably raw materials could be recycled for a while, however, and particularly decrepit areas could be torn down and turned into strip-mines in order to get more ore et al (stored in some buildings that are still standing for later).
Agriculture would primarily have to be supported via hydroponics and rooftop gardens. There isn't anything particularly far-fetched about that, although high levels of pollution might make it difficult to grow plants in a way that keeps them safe to eat. But if we assume that a highly-urbanized planet means that everyone can simply walk or take electric rail to wherever they're going, that seems a bit more plausible.
Animal farming would become much more difficult (but still not impossible), and I'd expect meat to become a luxury item with most people subsisting primarily on vegetables.
Electricity can certainly be generated with rooftop solar and wind power, as well as from biowaste (both from reclaimed methane and from good old-fashioned steam power from burning it). Much of the biowaste would also be usable as fertilizer for the gardens.
Oxygen sustainability almost certainly would become a problem; on Earth, most of our oxygen comes from rainforests and phytoplankton (i.e. the oceans), and deforestation would effectively cut our oxygen production in half. However, 21% of our atmosphere is oxygen and that's sustaining 7 billion people with plenty of room to spare, and only 0.04% is CO2. Even assuming that animal respiration is responsible for all that CO2 (which it isn't, as most of it comes from burning carbon-based fuels) that implies we could easily support several hundred billion, if not a few trillion, people with the oxygen in the air. (This doesn't mean that it would be particularly pleasant, of course, and we'd run out of other resources long before that, not to mention the huge impact that would have on the greenhouse effect.)
Now, there's about 57 million square miles of habitable Earth; assuming that the entire habitable surface of Earth had a population density similar to, say, Manhattan (which isn't the densest but it's not exactly sparse either!) we'd have a world population of around 4 trillion people, which would definitely be too much for our atmosphere to support (unless we were to somehow replace a lot of the nitrogen with oxygen, which would have some pretty major problems even if it were doable). Based on the above, with 4 trillion people, we'd expect the atmosphere to be something like 58% nitrogen, 21% oxygen, and 21% CO2. The greenhouse effect would be enormous, and everyone would constantly feel like they're suffocating (from the air, if not from the constant panic of being deep in an urban area that never ends).
Chances are, the added surface population would cause a massive algal bloom, making the oceans nothing but a giant oxygen factory. (This also accounts for where a lot of our waste would end up going.)
So, in short, it might be physically possible, but it certainly wouldn't be pleasant.
Simple answer:
Yes.
If the civilization is capable of interplanetary travel, definitely even plausible.
Long answer:
The only way you'd conceivably support the population would be similar to how large urbanized areas are today, except on a galactic scale: huge amounts of inbound and outbound traffic transporting raw and manufactured goods in and carrying waste out. This traffic would need to be capable of reaching other planets in relatively short time periods (days, maximum). Some amount of self-sustaining is possible utilizing near-future technologies in farming. Namely vertical farming, indoor farming and similar ideas. Although it may be easier/cheaper to import food as well depending on the interplanetary travel costs. Such industry would have a use for organic waste, such as sewage and reclaimed water, as well.
At some point, however, the planet would run out of things like iron, gold, silicon, etc. so those would certainly need to be supplied from off-world. Asteroid mining, gas giant refineries, and so on. Similarly, non-organic waste would likely be processed in some manner to recover usable materials (recycling: plastic, metal, glass, etc.) the remainder might be shipped offworld and dumped on non-habital worlds, dropped into stars, or packed into the mined out empty cores of asteroids (ah the future's garbage dumps...).
I think the society would have to be very technological advanced.
One big issue is energy. This would have to get out of the way by nuclear fusion or a dyson sphere.
Once there is energy, vertical farming can be used to create enough food and control atmosphere. Some process would be need to create huge amount of organic matter, using rock and specialized plants / bacteria.
With all that energy, a lot of heat would be produces, as energy usually ends up as heat. Energy efficiency is a must and some global cooling system probably as well. Controlling the atmosphere could help here.
The next thing are resources. High level recycling is a must, obviously. Usage of organic materials, long lasting products that are shared etc. Creating building material from compressed ground. Only walking-distance travelling, collaboration in virtual environments. Or just put them all in the Matrix ;)
I think that way, the city may actually be sustainable.
Computer systems to solve complex tasks are emerging. These would be necessary, to control a city of that size. The ruling party, that may or may not be private (if that separation would exist anyway), would be the one in control of the software system. The executive force would be controlled by the "AI" and therefore by the ruling party, kind of like a bureaucratic hierarchy without the hassle. Or an AI is already ruling on it's own anyway, which could also be the reason for such an omnicity.
In short: I think the key to a city like this is not import / export, but to gain total control of the planets resources, their transformation and the population.
Raw Material: This isn't a Problem at all. If you assume that you only build highbuildings like skyscrapers, this would be impossible without gathering resources from other planets. Not only due a deficit of materials like copper for power supply, but also i doubt that there's enaugh raw-material for concrete. Same Problem, when the world shall be completely covered in a mantling building.
But: urban sprawl doesn't mean "building skyscrapers". The largest port of an urban environment is covered with small or middle sized houses. Therefor, I think, is enaugh material on our planet (wich doesn't must be equivalent to yours.) If all the buildings are made from the same material, this could be problematic. But as you see in different parts of the world, different materials are used. Like claim, stones, wood and so on. Low-craftet materials does not mean, that a culture is barbaric.
Waste: You assume, that the way this world would handle their waste, is similar to ours.
But if a civilisation can manage it to only, or mostly, use near to 100% recyclable materials, the wasteproblem is not a problem any longer.
You would need:
Energy: Nuclear power, fission or fusion. Nuclear waste recycled or glassified and buried.
Electrical Food Chain:
Photosynthesis is just a way to knock electrons off atoms. In principle, it can be done with electricity alone. There's active research on this. Probably soon we'll have microbes that can produce sugar, fats etc from a current.
More simply synthesize high energy compounds and let microbes eat them.
Lastly, convert electricity to light, put plants under lights, stack as high as needed.
Regardless of method, food is produced in highly compact areas in quantity year round.
Carbon cycle remains closed.
Raw Materials No real changes, everything recycled simply because there is no place to put waste. Mines and the like are just in the basement.
Heat
The real problem with a planetary city is heat. Urban areas are already heat islands, always hotter than the surrounding natural terrain. Cover the entire planet and you'd heat the entire planet. Might actually cover the entire top in reflectors because really you don't need solar energy (solar panels are thousands of times less dense than nuclear power and they'd generate more waste heat per kilowatt.)
Presumably the planet starts as a terrestrial planet and has seas. Seas are the major heat conveyers on earth. They are powered by solar evaporation which changes density causes changes in density of seawater at top causing it to fall. If you build over the ocean you will stop the heat conveyers and the poles and the equator will have radically different temperatures which will produce storms. Likely, they will dump some waste heat back into the ocean to mimic solar effects and keep the conveyer going.
The real problem will be radiate enough heat into space. There is a phenomena called "heat pollution" that will occur when just the heat dumped into the atmosphere by technology starts to alter climate and drive storms.
Most likely a civilization advanced enough to build a planet city could figure out alternative radiators for the planet e.g. radiating lasers, magnetic radiators or just a bunch space elevator like structures that were just radiators carried huge amounts of heat from the ground to space.
Economics: To my mind, the major technical road block to a planetary city would be one of what pragmatic or economic forces would drive that many people to pay the enormous price of cramming that many people together?
Population will not grow as populations urbanize because children cost a great deal in urban areas and unlike under farming, produce no income until well into adulthood. Long before a planet gets paved over, population growth will be flat or even declining.
Dense urban cores, especially ones with skyscrapers, were justified by the need to cram lots of people close together so they could communicate with analog technology. Such dense cores are already technologically and economically obsolete, though not with out their lifestyle appeal for many.
We've long passed the stage where people had to be in New York to hit the big time. Major corporations are spread all over the world in urban areas big and small. It's as easy to send email around the world as in the same building.
I think the most likely scenario might be a completely non-economic reason to cram hundreds of billions of people on a single planet.
