I'm curious how popular executioners were, or unpopular. Did people hate them and what they stood for (the axe of the crown)?
Two articles which may answer your question, and which I will summarize:
Historically, executioner has been an "unclean" occupation. In ancient Rome, the carnifex, or public executioner, was not permitted to live within the city limits, but rather lived near the Sestertium, the area where slaves were punished and tortured.
In Spain the executioner was ritually untouchable, and executioners were made to paint their houses red and wear a symbol of the gallows on their clothing, as well as ritually confess to each killing in court.
Robin also lists in some detail the particular crimes of historically unsavory executioners, which I won't go into too deeply here. He does discredit the idea that the position was hereditary, stating, "...of the twenty-eight executioners mentioned in Laurence's A History of Capital Punishment only three had fathers who were executioners." He comes to the conclusion that many executioners for hire were unsavory individuals, perhaps unsurprising given the inclinations needed to kill professionally, but a few also wrote at length about their duty to be humane and attend the psychological needs of the victims.
Hornum discusses the historical origins of the professional executioner in Scandinavia at length, and discusses pay specifically. Evidently the Kingdom of Denmark in 1698 set a uniform rate for various executioner's services, including but not limited to burning, branding, breaking on the wheel, and beheading. He also discusses how Scandinavian executioners made money on various trades associated with their profession, including building and maintaining the gallows, disposing of the corpses of suicides, and in some cases even branching out into medicine (having the only source of bodies to learn anatomy from).
Hornum states that in the middle ages the position was considered extremely sinful, and that he carried a physical taint which others must avoid. He would have a dedicated plate and cup at the local inn so no one else would eat from the same dishes. He and his family had a separate pew at the church. Through the 17th and 18th century the attached stigma of the position decreased, but it also became less lucrative as there were fewer paying jobs that only a professional executioner would deign to handle.
Overall it seems that the position went from an "untouchable" one to a part of the civil service, and that national governments from the 19th century on made an effort to destigmatize the position, as it was sometimes difficult to find reputable executioners to represent the state.
TL;DR: The position was considered unclean and executioners were commonly shunned, but were the only providers of many vital services as burghs and villages grew more populous and organized through the middle ages. In later centuries (18th-19th) the position become less stigmatized but also less profitable.
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A little off topic, but, how are executioners treated now? How well are they paid?
It depends on when. You seem when speaking of Axemen to be covering earlier times, but we don't know a lot about executioners then. Most were fairly discrete about their jobs and in history, public executioners would often where caps to conceal their identity.
In England, execution was normally the responsibility of the local legal authority (the Sheriff) and then delegated to a person employed by the prison authority when required.
In 1868, Britain went away from public executions. Prison officials were forced to witness what was essentially death by strangulation at close quarters and didn't like it. William Marwood (1818-1883) developed the idea of the long drop, killing the prisoner instantly and was allowed to implement it in 1872. He was paid a retainer of £20 per year plus £10 per execution on being appointed the hangman to the Sheriffs of London and Middlesex in 1879. He died in 1884, so I guess he didn't do so well out of it.
James Berry (1852-1913) built on Marwood's ideas of the long drop, was probably the first hangman literate enough to write about his experiences so became a public name later on.
Albert Pierrepoint was a British Executioner from 1932 through to 1956 when he resigned (almost the last, but not quite). He hung 435 people in this time but ~200 of those were as a result of WW2. He followed in the footsteps of his uncle and father before him.
The job didn't pay that well (it was per execution rather than salaried), and he also worked in a grocer's store. Initially the pay was one and a half guineas per execution (about £98) plus the same again a week later if all went well. I think he also received travelling expenses. The fee increased when he was promoted to Chief Executioner, but it wasn't that much (about double). His identity was kept from the public until the hangings of WW2. After his involvement disposing of some very nasty war criminals, he was deemed to be a hero by the press.
Pierrepoint was unusual because we know a fair bit about him, from newspaper accounts that appeared about him (and after his resignation, with his help) and then later from his autobiography. Due to his position as (almost) the last executioner in the UK, he has also been the subject of biographies.
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