Depending on the timeframe of your apocalypse you, as the author, have certain liberties in explaining how anchorage became a major world player leading up to the incident.
JBH made a lot of good points about how to feed and power things so I'm just adding optional supplement to the notions.
Why anchorage? Because those who still had anything after the apocalypse went somewhere they thought they would be safe, and they brought what they could with them. Greedy pigs being greedy pigs, they began to restore what they once had using their greatest assets - that which others don't have. Opportunity tends to lend itself to those who have something to invest in the first place and the cheapest labor are people who can't otherwise feed themselves. So to rapidly rebuild, you become the provider for many.
Now, just because the world has been blown to hell doesn't mean humans dropped back to the dark ages. It's up to your own writing decisions to decide what tech survived. Communications and security are much easier when you're talking small scale, and a city like Anchorage is small compared to the rest of the world. So your communications grid is considered short range for the populace, and long range could fall back on the good ol fashioned radio waves.
Where does the food come from? At first, from anywhere you can manage. Assuming the nuclear insanity didn't happen on 100% of the surface of the earth all at the same time, those with the means moved their livestock and equipment with them in their great journey north. Livestock is a tough asset since generally speaking farms large enough to provide meat for a populace can't necessarily be underground residences without a lot more provisions that even the richest have to dispose of when there are cheaper and easier ways. Not to bring religion into this, but many people don't realize the major churches like the mormons keep massive seed repositories in undisclosed locations. No doubt when the feces hit the fan those in the know spread the word as carefully as they knew how that the great pilgrimage to the underground was under way. Any large group of people moving to one location is bound to attract attention and not everyone is willing to die to keep a secret. So, for the convenience of the story, someone got a hold of someone who had access to the right storage location and through badness a plenty you now have a generation's worth of seed deposits moving their way to anchorage where they use JBH's hydroculture to rapidly develop some remarkably pure organic vegetation from the now massacred mormon stockpile. It could be a year or so before such hydroponics facilities become viable and produce enough food to distribute. But you can always say those who had, or scavenged brought much with them. Plus, there's always those tubs of costco sized emergency rations that could be worked into the situation.
Much of your list is not a problem if you figure that not everyone who survives a nuclear apocalypse just defaults to mad max mode and goes rogue nomad with murder in their eyes. Police, hospitals, and all that are not foreign to anchorage and probably wouldn't even be disrupted if you chose to not allow it to be. Most of alaska knows more than everyone the concept of a septic line if sewage was an individual responsibility. Water can be cleaned from the gulf of alaska, desalinated, or whatever. Where there's a need and someone who can profit from it, there will be a way.
What's left on your list is little more than architecture of a society based on their means. Functional is then defined by how happy your population is, how safe, how determined, etc. Those who had the means came in, swindled their way into the high seats in the new world, and rule with the power of necessities with the blanket of impunity. That story is practically age old.