Is Satan inherently evil, or is he simply fulfilling the role that God gave him to attempt to lead people astray so that they can have the accomplishment of overcoming him? Or is he something else altogether?
God does not depends on Satan to test human. We all are equipped with rebellious nafs (ego, lower self). Satan, like many of us, slipped but unlike many of us, he refused to seek forgiveness, instead asked for respite and challenged God to mislead His creation. God gave him respite (now.. why? to test human? not necessarily) and said anyone who seek forgiveness He will forgive them.
Satan is God's creation. He was close to God but he slipped. He is very interesting character but nevertheless Quran declares him to be an open enemy to human. I do not think Muslims have any confusion or second opinion about Satan. Every year they stone him during Haj pilgrimage and he was eternally scorned for his deeds.
Satan is a Jinn (A type of creation like Human beings and angels) hence it is as insignificant as any other Human being in the eyes of Allah. There are human beings on this earth which are more powerful then others and have money, influence and power yet they can be either be believers(Mumineen) in Islam or disbelievers similar is the case of Satan(Iblis) who is a Jinn, he was one of the Jinns who was initially a believer Muslim, but his disobedience and arrogance made him evil and he became the cause of more evil:
Behold! We said to the Angels, 'Bow down to Adam': they bowed down except Iblis. He was One of the Jinns, and he broke the Command of his Lord....(The Noble Quran, 18:50)
"It is We Who Created you and gave you shape; then We bade the Angels bow down to Adam, and they bowed down; not so Iblis; he refused to be of those who bow down. (Allah) said: 'What prevented thee from bowing down when I commanded thee? He said: 'I am better than he: Thou didst create me from fire, and him from clay. (Allah) said: 'Get thee down from this: it is not for thee to be arrogant here: get out, for thou art of the meanest (of creatures). He said: 'Give me respite till the day they are raised up. (Allah) said: 'Be thou amongst those who have respite.' He said: 'Because thou hast thrown me out of the Way, lo! I will lie in wait for them on thy Straight Way: Then will I assault them from before them and behind them, from their right and their left: Nor wilt Thou find, in most of them, gratitude (of Thy mercies).' (Allah) said: 'Get out from this, disgraced and expelled. If any of them follow thee - Hell will I fill with you all. (The Noble Quran, 11-18)"
So to answer you question Satan was not evil initially , but the circumstances in which he was tested made him a disbeliever which led to his own downfall and a source of all evil. More over Satan can be also viewed as a synonym concept of evil and can be both among men and jinns:
Surah (Chapter) 114. 1. Say: I seek refuge with the Lord and Cherisher of Mankind, 2. The King (or Ruler) of Mankind, 3. The Allah (for judge) of Mankind,- 4. From the mischief of the Whisperer (of Evil), who withdraws (after his whisper),- 5. (The same) who whispers into the hearts of Mankind,- 6. Among Jinns and among men.
Even prophet Adam A.S disobeyed Allah but he did repent from the sins and the repentance was accepted, Satan on the other hand did not repent rather became more jealous of Adam and set out to misguide the progeny of Adam . Even among human beings those who reject faith of Islam are friends of Satan and evil, but they can cease to be satanic if they accept the God's message of Islam:
4:76 Those who believe fight in the cause of Allah, and those who reject Faith Fight in the cause of Evil: So fight ye against the friends of Satan: feeble indeed is the cunning of Satan.
I should point out that there is likely to be a confusion based on the usage of the word "Satan" for anyone not already familiar with the Islamic doctrine. "Satan" refers to two overlapping groups.
As the question uses "he" and "him," I assume that the assumption is that the word "Satan" always refers to the same being across the Torah, Bible, and Qur'an. This is incorrect. In the Qur'an, this being is referred to as Iblis. Like the other answers stated, he was instructed to bow down to humans, and refused. Subsequently, due to his anger at being rejected by Allah, he has chosen to lead people astray from Allah's desire for them. Iblis' actions are his own, caused by his pain at his just punishment.
The word "Satan" in Islam also refers to a class of jinn called the shaitan. These beings are actively malicious, because they have never heard the Qur'an, which would teach them to be good. They can be repelled by readings from the Qur'an. These do not convert them to good, suggesting they understand the effects the Recitation would have on them, and seek to depart when they hear it. Iblis is the leader of the shaitan, and is counted as one, leading to some confusion.
We believe that Shaytan/Satan to be mankind's greatest enemy. Shaytaan has been granted life until the last day, and his goal is to mislead mankind, those who hold strong faith and do good we believe Shyataan will never get them.
And [mention] when We said to the angels, "Prostrate to Adam," and they prostrated, except for Iblees. He was of the jinn and departed from the command of his Lord. Then will you take him and his descendants as allies other than Me while they are enemies to you? Wretched it is for the wrongdoers as an exchange.
Surat Al Kahf Ayah 50
Lastly I will like to end mentioning the 62 verse of Surat Al Isra'
[Iblees] said, "Do You see this one whom You have honored above me? If You delay me until the Day of Resurrection, I will surely destroy his descendants, except for a few."
I ask Allah for forgiveness of my shortcomings and mistake, and nay He guide us all.
Is Satan inherently evil?
Let's define evil...
Evil: immoral, cruel, or very unpleasant (Cambridge Dictionary)
Evil: something that is very bad and harmful (Cambridge Dictionary)
I think the 2nd definition is good in regards to Mr. Satan. Not, pretty much the 1st one. It is man who fulfills the 1st definition under the influence of Mr. Satan possessing the 2nd definition. So, it depends on how you define evil.
Let's define inherent...
inherent: existing as a natural or basic part of something (Cambridge Dictionary)
So, in his basic form, Mr. Satan (a general name given to the descendants of Iblis i.e. the main Villain at the time of Adam (pbuh)) belongs to the class of Jinn, who by their very basic nature are given free will. Hence, by misusing the free will, Mr. Iblis and his descendants chose to be evil i.e. by deceiving people, and God let that be because Mr. Iblis requested so.
is he simply fulfilling the role that God gave him to attempt to lead people astray so that they can have the accomplishment of overcoming him?
May the creator guide us all.