I checked this on WORD DETECTIVE and got no answer. What is the entomology of the word demonstrate? Is it evil? Should it be avoided?
I think you have a bug in your question.
For “just words” you are often better off checking a dictionary than the Word Detective or similar sites that specialize in odd phrases.
From the Merriam-Webster site:
de- is a Latin prefix indicating a source or “from”, so demonstrare was to show something from its source or to show how something came to be.
Demon comes from the Greek daimon and remains in English as dæmon (or daemon) in some constructions.
So the word is about the removal of monsters rather than anything to do with (L) daemons.
Why the demon in demonstrate?
Coincidence, like the “bat” in “bath” or the “harm” in “harmony.”
At the risk of having been whooshed,
no, the word is about showing something, particularly how it works or where it came from. According to the etymology you provided, demonstratus was already cognate with demonstrate even before the word came out of Latin.
The “monster” connection is to the root form, “to show,” and led to the English word monster, meaning a creature that was displayed as an oddity. The similar meaning of “to show (bad things in the divinations)” added a layer of expected evil to the word “monster,” but demonstrate had already gotten its own definition (in Latin) before all those side shows (SORRY) with the root definition developed.
Another form of monstrare led to the name of the object in which Catholics place the consecrated Host to be displayed at various rites, the monstrance.
Or the “fun” in “dysfunctional.”
Or funeral.
Okay, that gets a :).
Interesting fact - in Afrikaans, the word for “sample” (like, blood sample, or water sample) is “monster” (pronounced somewhat like “mawn-ster” rather than “mon-ster”) - I wonder if it’s etymologically related?
Demon may be in demonstrate, but as you know, the devil’s in the details. 
Tom, I think you’re wrong about how “monster” came into english. Monsters weren’t named because they were displayed as oddities, rather monsters were divine omens. So when God was pissed you’d have comets, rains of blood, birth of two-headed calves, dogs and cats living together. Deformed animals were signs. Eventually the word came to mean only deformed animals, then to the current meaning.
Let us not dwell on the town of Scunthorpe.
What is your objection to Thor?
I read once that “demon” originally meant “teacher”, which would jibe with “to show”, mentioned above.
I remember a strident, screeching, ball-busting feminist (I am not saying all feminsits are that, just that I met one who was) who insisted that men are naturally programmed to tyannize and dominate, which is why we have words like “manipulate”.
She seemed completely unconvinced when I pointed out that the “man” in that word was from the Latin “manus” and had NO conection with the word for male humans, which would have been “homo” in Latin. Then again, I guess listning to anything a man had to say was asking a lot of her.
So always beware of “words” that occur concidentally in another word. Or as an old joke puts it: "Dolly Parton put the ‘cunt’ in ‘Country and Westrn’. " 
You have to see what was left on de plate, if you want to know how much de monster ate.
Or the ‘fun’ in Fundamentalist Dogma
If you’re worried about that one, I hope you’re also avoiding using the word “hello.”
I do apologise: I did not mean to insult your deity, merely to point out that “pe” could sound rude one’s sensitive maiden aunts.
I’ve heard (from my close personal friend, Al Yankovic) that Jerry Springer puts the “sin” in “syndication”.