Well, they have a technology called “replicators,” which allows them to manufacture almost any form of matter out of energy. This might not seem like a very useful technology, since if I remember my high school physics correctly, it would take the power of a thousand exploding suns to create enough matter for a decent New York strip steak and a side order of mashed potatoes
This is not a very hard problem. If you need energy to make some mass, you break up some other mass to make energy. Plenty of universal copiers in SciFi have an "in" hopper, which you fill with whatever sand or dirt you happen to have. Inside the machine that mass is converted to energy and rebuilt into atoms that you need.
Is the Federation lucky enough to have an adequate number of people who find self-fulfillment by volunteering to work in dingy hell-holes, digging up those precious crystals?
Yes, this is one of major, carefully ignored problems with Communism. Everyone will want to be an artist and singer, but very few would want to operate robots that clean sewers. It is unreasonable to postulate that such jobs will not be required. They are required even in the Star Trek universe. They will certainly be required in the real world. No robot will be able to get into some stinky place and figure out, all by itself, what happened and how to resolve the problem. (Even if such a robot is created, like Data, he would not be particularly happy to do the work - and you don't want to face 100,000 unhappy androids; they might discover that it's the humans who are the ultimate reason why the sewer is blocked.)
He might indeed have been cooking because he enjoyed it and found it fulfilling, but what about the people waiting tables in his restaurant?
This is another example of a job that is laborious, hard on people, and not very much fulfilling. Working with people is always hard. The owner would have to resort to using robots as servers. Still, he would have to find someone to work as a maître d'hôtel - at least to know what his customers want to order.
Or is the future Earth filled with layabouts who just watch holographic game shows and replicate Hot Pockets all day?
Pretty much that's the official position of Communists. "If you don't want to work, you don't have to." I can still imagine some middle aged guys continuing to work out of habit that they worked out in old, bad days of Capitalism. However what in the world would ever force teenagers to abandon their computer games, their relations with opposite sex, and their tribal instincts to go somewhere every weekday and work for several hours? Glamorous work, like racing car driver, Starfleet (but not Red Shirt,) perhaps art - those will be filled for various reasons. But most of work is not glamorous at all. Robot techs will be the primary occupation of those who work. And if the broken robot was a city bus, you have to work on it in whatever weather is out there. If the broken robot was cleaning sewage... I guess you will go there as well. And who among humans, may I ask, will volunteer to do such work? Such a society has to change humans into some sort of hive mind, like ants or bees.
How do you get a table at his restaurant? Is there a four-hundred-year waiting list, the way “free” medicine is rationed in socialist countries? All the free matter and energy in the universe can’t change the fact there’s only one Sisko Senior, and he’s only got two hands to cook with.
Such place won't exist because no sane person can volunteer to cook for hundreds of patrons every single day. Today an amateur cook makes a dinner now and then for a few invited people, and it's a lot of work already. One person in a whole restaurant cannot cook enough food to sustain a restaurant of any reasonable size (say, more than one table.) But even if that is managed somehow, by using a single dish of the day, or a robot helper, still nobody would want to spend his life cooking. Today people work primarily because they are motivated by necessity - they can sell their labor on weekdays, and then on weekends they can buy something else. Most people who work menial jobs would gladly abandon work if only they could. Their life, their time with children and grandchildren, their appreciation of nature, their enjoyment of existence cannot be made in a replicator - it's a very finite product, and every minute of work distracts you from watching the clouds (as an example.)
How are the services of the top doctors assigned? Are they exclusively assigned to take care of high-ranking military officers and Federation politicians?
Doctors are yet another example of limited human resources. If you want a human doctor, you need a motivation for that human doctor to exist. Some will be motivated enough by their desire to heal people. Perhaps there will be quite a few of those among doctors. But will the society have enough human nurses? It's not one of those glamorous jobs either.
Politicians are yet another issue. Communism is not supposed to have a government. People just magically do what is right, all the time and in every circumstance. SciFi suggests that there would be some Councils that gently guide the society... but if the members of the society are free, they have every right to ignore those recommendations. The government cannot function unless it can somehow force the people to obey. (This is why theoretical Communists dislike governments.) This is where their deus ex machina comes handy: in the Communist society everyone - magically - willingly obeys the decisions of those Councils. Even if those decisions kill them, and they know it. There was an original Star Trek episode about two planets that simulated attacks on each other, but killed their "victims" for real. That episode really pushed the limits of belief. It's not in human behavior to meekly go into disintegration chambers.
Communists are painting an appealing future, but it is always full of holes like these. Sociologists do not know how to patch them up. We have to invent those methods pretty soon, as automation (and cheap foreign labor, which is equivalent to automation) is sending many US workers into involuntary retirement. Why does the USA need a metalworker to make a pot if the whole thing can be obtained, already made, from China in exchange for a number in a computer somewhere deep within Federal Reserve? Who would need workers if every Wal-Mart has a replicator in the basement? Who would have money to buy anything from that Wal-Mart, if nobody works? If everyone wants stuff for free, why that Wal-Mart should even exist, as it takes nonzero human labor to run replicators? Somewhere in the chain, near raw resources, or near the energy (those Dilithium crystals,) or near antimatter reactors you have to have people who work, as opposed to everyone else who does not work because there are no jobs for them. What will make those workers spend their life doing dangerous work if they don't have to? Even if you get a starry-eyed teenager interested, his interest will disappear after a few boring shifts, or after a few weeks of crawling through the bowels of the reactor, with tools in his teeth, while his friends relax at a beach and chase girls. That behavior would be normal for an ant, but not for a human. Chances are that humans, as we know them, simply cannot form a Communist society. It's far more likely that they will form an Anarchist one, just as they have done so for many centuries of written history.
