How did the police department survive, and why were they allowed to?

I do remember several scenes (quick ones) and off-the cuff explanations that they were receiving supplies through the man-hole that Blake's partner emerges out of near the climax as well as through the destroyed football stadium (at one point you see a whole pallet of supplies being hoisted down from...somewhere).

While vague, the audience is expected to assume the supplies they were receiving were enough not only for the citizens of Gotham, but enough to deposit down to the police's prison underground, allowing them proper sustenance.

The only logical explanation that Bane kept them alive aside from arrogance that no one would attempt to free them; is for Bruce or even Gordon to know the full extent of their failure to the city, having the police suffer down in the confined depths. If memory serves, however, Bruce doesn't know about the plot to trap the police until he finally returns to the city, so it is more likely this is Gordon's punishment.

It might be an attempt to humiliate them: it seemed to me that Bane tried to reverse everything(the rich living on the streets, the righteous being judged).

So in this situation, having the police trapped in a hole (maybe an analogy to the "hole"/isolation where prisoners are sent for very bad behavior) and fed by the prisoners might make sense. This would also explain why they were shaven: they were offered the same treatment criminals in a prison would get.

Still, killing all the policemen at once (crushed like bugs) would have been, in my opinion, more tragic and a lot safer for him.

But, as we have seen, Bane has a sense of drama, and all his actions have to be somehow theatrical (he only blows up the stadium after the anthem is sung, keeps the bomb in a truck and has it driving through the whole city in order to remind the people of his power, and the examples could go on), so aspects like safety or big sudden tragedies are not very familiar to Bane. He said it himself that he finds it more enjoyable to let the people have a hope and then crush it, repeating the process until the citizens are truly broken (the same strategy is he trying to apply to Batman).

Their survival is covered in the film's screenplay. In short, Bane is feeding them.

INT. SEWERS – DAY 207 Hundreds of Police living underground. Dividing up supplies lowered by ropes … Ross opens a pack of supplies.

BANE: (voice-over) But the police will live, until they are ready to serve true justice …

There's also a brief mention in the novelisation about them surviving on "vermin and scraps"

Blake hoped Ross and the other entombed cops were collecting the icy water. Nearly three months had passed since Bane had sprung his trap, and the buried officers had been living on scraps and captured vermin ever since. It was a wonder that the buried officers hadn’t yet completely given up hope.

As to the numbers, it was certainly into the thousands, according to the script and script notes:

REPORTER: We’re seeing literally thousands of police heading into the sewers –

...

SWATs react – the tunnel roof behind collapses … Chunks of concrete structure drop – thousands of police throughout different tunnels are trapped …

There were, I think, around three thousand police, almost all the GPD's active duty force. They were there the full five months (or however long). And they were being fed, probably by Bane.

It could be they looked good because whatever GPD commanders and NCOs were down there decided that it would be bad for morale to go around looking like unshaved, shaggy-haired scarecrows dressed in rags. After all, they didn't have much else to do down there! Allowing them a few razors and clean uniforms affords them some hope that Bane will then crush for... reasons.

Bear in mind Bane's motivations simply weren't rational. He said he wanted them to join the revolution, something, something, who knows? Something more about fear and hope. Etc. He was on a suicide mission anyways, carrying out Ra's al Gul's plan. If simply destroying Gotham was the plan then they could have done that as soon as they had the bomb, with Batman/Wayne watching it happen on TV from prison.

One tentative reason for keeping the police alive is this is part of the ruse of wanting to rule Gotham instead of destroying it. That is, as long as the police are being kept alive by Bane the government might still think it has time to put in motion its own plans and/or Bane is capable of negotiating in good faith. If the police are slaughtered they might realise his plan is to destroy Gotham all along and put into motion more active plans. (Fun fact, as the bomb is being driven around in trucks could easily be taken out by an EMP, or for that matter a drone-launched Hellfire missile or precision-guided weapon without detonating. Heck, you could do the latter even with a military nuclear device, but I digress.)

Two-A, following from this, the lives of the police are bargaining chips to be used in some hypothetical future negotiation with the government, again, as part of Bane's overall ruse. By keeping them alive he appears merciful, even reasonable to the people of Gotham and the US government. Dead, they are worth nothing.

Two-B, The people of Gotham seem to be keeping their heads down. Not joining Bane wholeheartedly but not offering large-scale resistance, either. The people of Gotham are offered some hope that Bane is not a complete monster and is capable of sparing his enemies in some fashion. If Bane simply slaughters the police the people of Gotham might realise their only hope is to resist because Bane is a madman and they have nothing to lose.

Three, their lives are essentially being held hostage for the government's good behavior. The government already knows Bane is bluffing somewhat; sending in Special Forces resulted in their death but did not trigger a maximum response of Gotham's destruction. This is also a real-life problem with nuclear war and "Mutual Assured Destruction": a massive strike might trigger a maximum response, but what if instead of a thousand incoming ICBMs there's only twelve? Or Three? If your only creditable option is to burn everything down to the ground you have a difficult choice! The outcome of this was a set of "Limited Nuclear Options" (LNOs), limited retaliatory strikes short of all-out retaliation. In the same fashion by keeping the GPD officers alive but trapped Bane has a threat in his pocket that is a step short of escalating to thermonuclear oblivion. For example, if the government attempts to infiltrate Navy SEALS into Gotham Bane can send in his goons to kill a thousand police in retaliation instead of triggering Gotham's complete destruction. This seems like doing things the hard way because the police are still armed and capable of resisting but it's clear Bane doesn't value the lives of his men, who will also give their lives for him. Again, this is only in service of keeping his main ruse going.

The reason Talia and Bane are in Gotham City is to simply destroy it,and fulfill Ra's al Gul's plan of wiping out the the most crime ridden and rotten place on earth. Ra's believed that Gotham was too far out for redemption.That's why Bane buried the cops under ground so that when the neutron bomb went off the city's guardians "the cops" would still be there afterwards. Bane didn't hate the cops nor was he sadistic, he just wanted to pave way for a new and brighter Gotham and to teach humanity a lesson,and what better way than to use a neutron bomb. A bomb that does little damage to environment and buildings but is extremely lethal to organic life "humans". The choice of the bomb itself shows Bane and Talia's motives clearly.

I'm sure they could have easily acquired and used an old fashioned uranium or hydrogen bomb,but if they did there would be no Gotham left for them to realize their dreams.

Gotham's decadence was too much and it was time for it to die,it was beyond saving...those were Ra's al Gul's words and wishes.