How frequent were murders by Mormon Danites in the 1850s?

I notice that you do not cite any sources for the existence and activities of Mormon Danites, however after some research I think I can provide a reasonable answer to your questions.

It seems that the question of whether Mormon Danites actually committed any murders at all depends largely on whether you believe the evidence given by Mormon dissident, Dr Sampson Avard, at the trial of Joseph Smith following the 1838 Mormon War in Missouri.

There certainly seems to be very little evidence for activity by a Danite sect within the Mormon community after 1838. The Mormon Encyclopedia essentially states that claims of Danite activity after 1838 were based on Avard's account and simply

provided a ready explanation for anyone who wanted to believe the worst

In the 1870s, two former Mormons, Ann Eliza Young and Fanny Stenhouse wrote "exposés" of Mormonism. Both are available on Archive.org:

Ann Eliza Young - Wife No. 19, or The Story of a Life in Bondage ; Being a Complete Exposé of Mormonism, and Revealing the Sorrows, Sacrifices and Sufferings of Women in Polygamy

Fanny Stenhouse - Tell it All: The Story of a Life Experience in Mormonism

Both claimed that Danites were still active and discreetly murdering Mormon dissenters and outsiders who were seen to be a threat to Brigham Young. The problem with these accounts is that both rely entirely on hearsay and rumour, and that neither offered any evidence to support their claims. As an example, on page 170 of her book, Fanny Stenhouse reported a conversation she had had with a lady named Mary Burton. She records Mary Bruton's side of the conversation thus:

"... for I have heard from people who ought to know, that since the Saints have been in Salt Lake Valley the same things have been done; only now that speak of these men as "Danites" and "Avenging Angels." People say that those who are dissatisfied and want to leave Zion, almost always are killed after they set out, by the indians, and they dare not say boldly who they believe those "Indians" are. The too one lady told me that she had heard from her sister that not only were apostates killed in a mysterious way by Indians or some one else, but that many people were "missing," or else found murdered, who were only suspected of being very weak in the faith"

Despite the claims of Danite activities in her book, Ann Eliza Young did observe that:

Joseph Smith always denied that he had in any way authorised the formation of the Danite bands; and, in fact, in public he repeatedly repudiated both them and their deeds of violence.

(p48)

Although hardly authoritative, the Wikipedia page notes that:

Not a single "murder" was reported during that time to support these allegations, not to mention evidences which would support allegations of multiple homicides.

and that

To date, both LDS and non-LDS researchers have failed to produce any evidence providing (even remotely) the support of such claims, leading to the position among scholars of Mormon history that the claims of Ms. Young and Ms. Stenhouse were wrong.

One of the more notorious incidents claimed to be the work of "Danite bands" was the 1857 Mountain Meadows Massacre. John D Lee was arrested and tried for leading the massacre. Although he never denied his own complicity in the massacre, Lee claimed he had not personally killed anyone. In court he claimed that he had been a reluctant participant and was later used as a scapegoat to draw attention away from other Mormon leaders who had been involved.

Throughout two trials (first in 1874 (which ended inconclusively with a hung jury) and then in 1877 where Lee was convicted), Lee maintained that Brigham Young had no knowledge of the event until after it happened.

Now, in the book The Life and Confessions of the late Mormon bishop, John D. Lee, published after his execution by firing squad at Mountain Meadows on March 23, 1877, Lee (or an editor) wrote:

"I have always believed, since that day, that General George A. Smith was then visiting southern Utah to prepare the people for the work of exterminating Captain Fancher's train of emigrants, and I now believe that he was sent for that purpose by the direct command of Brigham Young."

(p225)

The problem here is that this statement was published posthumously, and is clearly at odds with his earlier claim that Young had no knowledge of the event until after it happened, which obviously raises the possibility that it was added later by an editor or publisher.

Despite this, Lee's book has been used extensively by modern authors to demonstrate the activities of Mormon Danite bands. An example of this is American Massacre by Sally Denton. On her website, Denton describes herself as

"Investigative Reporter and Author"

I can't claim to be familiar with any of Sally Denton's work personally, but I did find a number of critical reviews of this particular book (for example, this piece by Robert H. Briggs). The main criticisms of American Massacre seem to centre around her reliance upon some controversial psychoanalytical methods by Fawn M. Brodie and Robert D. Anderson (perhaps others may be better able to expand on the strengths and weaknesses of these methods), and upon the unreliable posthumous testimony of John D. Lee, discussed above.

Given the paucity of contemporary evidence for the activities of the "Danites" in the 1850s, and the unreliable nature of such evidence as does survive, it is perhaps unsurprising that these stories appear to have been a popular topic for pseudo-histories over the years. However, I think that Leonard J. Arrington summarises the current position best in his biography of Brigham Young:

As a part of the law enforcement system of the State of Deseret, Brigham Young had created a small force of Minute Men who were prepared to leave at a moment's notice to pursue Indian or white raiders in order to recoup stolen cattle or horses. So efficient and dedicate were these young men that they began to take on a sinister aspect to those who observed the workings of the Mormon system from afar. Taking some licence from the short-lived nonofficial Mormon vigilante group in Missouri, they were sometimes referred to as Danites or Destroying Angels. They played, and continued to play, a major role in western fiction, and many readers have imagined Brigham as a military dictator with a personal army of avengers who carried out his orders to capture, torture, and kill people who crossed him. But that Minute Men were anything more than a group willing to undertake arduous labors for their governor and church president has never been demonstrated.

Leonard J. Arrington, 2012, Brigham Young, Doubleday, p250

So, to answer your question:

Did Mormon Danites actually exist?

Very possibly, although whether Joseph Smith or Brigham Young actually knew of their activities is a matter of dispute. There is certainly compelling evidence that many people believed that they existed, and that they were afraid of them.

How common were the Danites? Were there Danite divisions in every ward and stake in Zion?

We have no way of knowing. Again, there is good evidence that people believed that Danites existed and that they were widespread in Utah, but we lack the evidence to determine whether those beliefs were based on fact.

Were Danite murders rare events committed by vigilantes or was Brigham Young's Utah basically a Fascist state, with the Danites playing the role of the institutionalized secret police?

Although we do have evidence of murders by vigilantes (the trial of John D. Lee, for example), I have not been able to find any credible evidence that any were carried out by "Danite groups" associated with Brigham Young.

There certainly doesn't seem to be any evidence that Brigham Young's Utah was a Fascist state, or that Danites played the role of his institutionalized secret police.