I've been reading a great deal lately about the alchemists out of mere curiosity. Many seemed to discover pretty ideas and were genuinely brilliant: John Dee, Ramon Llull, Issac Newton. Others, more occult than science seemed more geared on necromancy and witchcraft. It seems that in the 19th century alchemy and occult practices took a turn for the less scientific and "Christian" and more to the Kabbalistic and pagan, resulting in people such as Aleister Crowley. What did most people in Medieval times and the early modern era think about alchemy and occult practices? Where many prosecuted or killed in witch trials and such? What exactly was the relationship between alchemy and the occult? To what degree is the occult related to Freemasonry and the Rosicrucians? I read that Pope John XXII was an alchemist and then that he forbade it? Did alchemy and occult practices come from monks who studied ancient literature and perhaps weren't exactly keen on Christian scriptures? What about the Greek and Egyptian factor a la Hermes the Thrice Great?
I know that's a whole lot. Any one of those answers at a time will be good. I have always been fascinated by this stuff and was curious as to its extents in real life.
Oh, boy. I'm literally writing two books on this very topic.
I'm a co-host of the history of alchemy podcast, and we talk about this all the time. We did an episode on everyone you mentioned (including Rosicrucians) except Aleister Crowley.
Okay, so to your question:
Alchemy's birth is the same place and time (4th century Alexandria) as some philosophies like hermeticism, gnosticism, neoplatonism, etc..
So it gets mixed in with those, since early alchemists (like Zosimos of Panopolis, let's say) were also often gnostics, hermeticists (like Zosimos) and neoplatonists.
Then the tradition moves to the Middle East. There it has a strong medical component. Many alchemists were also physicians. And this tradition continues when it comes to Europe in the middle ages, and this medical tradition continues all the way to the court physicians (I mention many on the show) in the Early Modern period.
But one of the books I'm writing is about the scientific contributions.
There are many, but let me give you some examples that cross both occult and science: the discovery of phsosphorous - by hennig brand - was done using an alchemical recipe with aspects of sympathetic magic.
The discovery of dyes like prussian blue and some red scarlet were discovered by tinkering alchemists on the search for the philosopher's stone.
Control systems like thermometers, self regulating furnaces, etc. were done by alchemists like Dippel.
But alchemy lasted over a millenium.
So we have alchemists (and people at large) that thought alchemy was natural, or supernatural. If natural, that humans can attain the philosophers' stone. Others thought humans cannot.
If supernatural, some thought it was with the help of angels (Kelley and Dee) or God (many, but even Isaac Newton), and others thought it was with the help of demons (the Inquisition)
etc., etc., etc...
Okay. then the next part of your question: Rosicrucians. John Dee influenced their founders. Christina of Sweden was said to be involved or interested. Rosicrucians themselves mention many alchemists that had nothing to do with Rosicrucianism.
Next part of your question: Alchemy was intertwined with hermeticism and neoplatonism in Alexandria. But then it made it's way East to Persian and Arabic countries through Nestorian Christians and Jewish translators. Those Arabs commented on these alchemical works (as well as works by Aristotle, Plato, Galen, etc)... many trying to make it fit with Islam.
These commentaries made their way back to Europe. Through folk like Ramon Llull, the Crusades, Byzantian alchemists and philosophers, etc.. it's then monks (mostly) speaking Latin that translate these texts, and again comment on these texts. Now Arabic and Persian alchemists add to the mix, but again these first translators are impressed with what they read and try to make this "fit" with Christianity.
We have popes banning alchemy, we have popes banning alchemy, but then secretly ordering their own experiments (we see the same with kings, dukes, and even emperors) and we see pope alchemists, emperor alchemists, queen alchemists, etc.
So it depends on the specific pope, alchemist, emperor, queen, king, etc. that you're talking about.
Do you have some follow up questions? I could talk about this all day. Feel free to ask about much more specific examples. Your question(s) is super broad, so, so is my answer(s)
Also, some parts of this (if you're curious) I've answered elsewhere. Just take a look at: http://www.reddit.com/r/AskHistorians/wiki/profiles/bemonk
and scroll down to the AMA's and Alchemy sections.