Great question, I love me some, dead religions. Well, dead might not quite be the right word: apparently the rise of ISIS has led to a resurgence in Zoroastrian belief in some areas of the Middle East.

Zoroastrianism is one of the world's oldest religions because it predates the creation of Christianity by somewhere between 600 to 1000 years. (Duncan Greenless argues for a date between 1200-900 BC for the life of Zoroaster, but more commonly accepted is a date around 650-600 BC, as I understand it; Wikipedia places him as "traditionally dated between 628-551 BC." Basically, no one can quite agree.)

Anyway, to translate Wikipedia out of pretentious academic and into normal English, what "cosmogonic dualism and eschatological monotheism" means is that Zoroastrians believed the world was the battlefield of two gods, Ohrmazd (Good) and Ahriman (Evil). A Zoroastrian's job was to aid the good god while thwarting the evil one. "Good thoughts, good words, good deeds," is a good approximation of their belief system; slightly tangential, but a nice fun fact: Freddie Mercury, of Queen fame, was a Zoroastrian.

These next beliefs are a little tougher to ascribe for certain to Zoroaster, as they may have been slipped into the religion later to make it more palatable and easy to spread, but for examples of what deeds the Zoroastrians thought were good and part of their responsibility as the agents of a good god were the creation of fires (the "magi" referred to in the Quran are a reference to this; Kerdir, who I will mention later, in an inscription takes great pride in lighting fires across the land), and the killing of creatures like serpents and scorpions, which were considered agents of Ahriman.

Dualism refers to the fact that it ascribed creation to two gods. Greenlees recounts some versions of their creation myth where Ohrmazd and Ahriman are brothers, both of whom descended from Ahura Mazda. After praying for one son, he created the smelly, disgusting Ahriman, prayed again, got things right, and came up with Ohrmazd. Unfortunately, I can't find another source for these myths online and most other places refer to Ohrmazd as another name for Ahura Mazda, so Greenlees might just be wrong there, or gotten a little carried away. Some, particularly Greenless I believe, argue that since victory is inevitably ascribed to Ahriman and the light, Zoroastrianism is not a "true" dualism, where either god could emerge victorious, but that's, in my opinion, getting a little pedantic.

As far as its relation to Christianity: many consider Zoroastrianism to be the world's first monotheism, which came to inspire Jewish tribes, and later Christians when they finally split off several decades after the death of Jesus.

If you want an example of a Christian sect that drew rather heavily on Zoroastrian beliefs and blended them with Buddhist and Christian ideas for a syncretistic religion, take a look at Manichaeism, which also sprang up in the Middle East (roughly modern-day Iran) and predates Islam by roughly four hundred years. Today considered a heresy, reading their documents shows pretty clearly that Manichaeans viewed themselves as Christians; based on the extensive persecution they faced, the fact that they were driven into extinction, and that their holy texts were destroyed so extensively that we don't even have a copy of the Arzhang, it's fair to say the Orthodoxy did not agree, though it wasn't just Christianity which frowned on the Manichees: they arose in Iran and Kerdir, then effectively the leader of the Zoroastrian community, lobbied hard against Mani, leading to him being bound in chains and left to die under the rule of Bahram I in the Sassanian Empire.

Within the Manichaean cosmology, Light and Dark are two forces, the Light purely spiritual and the Dark physical. The lust of Dark for Light leads to an invasion of the Land of Light, where the demos of Dark glut themselves on Light. These demons subsequently turn on and consume one another until the two left at the end, male and female, at the end mate and produce Adam and Eve.

Our world, in this belief structure, is the creation of the King of Light, an alchemical tool meant to refine the Light out of the Dark so that it might be returned and ferried back to the Light realm by the sun and moon. Humanity's task is to liberate the divine sparks of Light trapped within us (to give an idea of how this impacted their beliefs in a practical way, Manichaeans frowned on sex and reproduction as it was believed to further the goals of the Dark by diluting and trapping Light in more physical bodies; they also had a somewhat bizarre diet that frowned on meat and thought melons contained more light and should be a staple of one's diet, which Augustine mocks them for after departing). Once the creation of a new Paradise is completed by an architect aspect of God, the current, physical world collapses in like a cage to imprison the darkness.

This is radically different than the Zoroastrian belief structure, but leans heavily on its dualism of good vs. evil (albeit taken to an extreme) as opposed to earlier 'pagan' religions, which more often attempted to unify man in order to have someone come to terms with nature and see ourselves as a part of the world's natural cycle; Freud and Jung both have similarish ideas, believing that repressing our 'bad sides' as Manichaeans propose rather than working to understand them leads to deeper psychological problems). Put more simply: one sees humanity as a part of nature (and I think Zoroastrianism still falls into this realm, mostly) whereas the other sees humanity as distinctly separate and alienated from it. Saint Augustine, one of the Fathers of the Church, was a Manichaean hearer (rather than an Elect, who had to take on more stringent rules to live life) for a long period of time, and several authors believe that it was his time as a Hearer that led him to propound a worldview with humans as distinct from nature.

As far as a direct link or influence on Islam, that's a subject I'm not particularly well-informed on, though, as mentioned earlier, the "Magi" referred to in the Quran are the Zoroastrians. As I understand it, Islam drew more heavily on the influence of Christianity and Judaism in the area than Zoroastrianism, which I would guess was consigned to the realm of the jahiliyyah or Time of Ignorance. Since monotheism derives from Zoroastrianism, it can be argued it influenced Islam with this concept, but I believe Muhammad was more reacting to the Abrahamic faiths than that of Zoroaster, trying to return to tribal and communal values rather than the commercial, individualistic ones his tribe had begun to embrace within two generations of his life. As far as why it hasn't spread to Europe.... great question, and I'll have to leave that for someone else because that steps outside of my field of knowledge.

Sources:

Paul Kriwaczek, In Search of Zarathustra (I would recommend this as a good starting book on the subject)

Duncan Greenlees, The Gospel of Zarathushtra (would not recommend this as heavily; tough to find, a bit on the pricey side, and not incredibly readable, if you ask me)

Duncan Greenlees, The Gospel of the Prophet Mani

Geo Widengren, Mani and Manichaeism

On ISIS and the resurgence of Zoroastrianism: http://www.thedailybeast.com/articles/2015/05/31/fed-up-with-islam-and-sectarianism-some-iraqis-embrace-zoroastrianism.html (Didn't realize this was the Daily Beast, which I believe has questionable credibility).

I have yet to read the following two books myself, so I hesitate to recommend them, but they are the next two I plan to pick up to deepen my understanding of Zoroastrianism: Jenny Rose's Zoroastrianism: An Introduction and John W. Waterhouse's Zoroastrianism. I'd say Kriwaczek is your best bet for a first book, then either these or Greenlees; for the first section of his book, at least, before he dives into Zoroastrian scriptures, Greenlees does do a really good job.