The story is "The Thirteenth Floor" by Frank Gruber, which you can read online here, and a bibliography showing various magazines and anthologies it's been published in can be found here. The first line:
The motto of The Bonanza Store was: "If The Bonanza Hasn't Got It, It Isn't."
The story also features a character with the name "Richard Javelin". He is ...
It's "Manners of the Age" by H. B. Fyfe. The story is available online here or here.
The red tennis robot scooted desperately across the court, its four
wide-set wheels squealing.
The anthology mentioned in the question is the generically named "Science Fiction Stories".
Could this be the story "Hide and Seek" by Arthur C. Clarke?
Most of the details match; it's told as a story within the short story, a lone astronaut avoids capture by a large cruiser. The major difference is that it takes place on Mars' moon Phobos and not an asteroid.
I took a look for what anthology you might have read it in, but appears to have been ...
"A Small Kindness", a 1983 short story by Ben Bova. Any of these covers look familiar? (I don't have any of those anthologies, so I don't know, but none of them looks like a "dark fantasy" anthology.)
. . . an assassin is contracted to kill a politician,
Terminate Rungawa. That was his mission. Kill him and make it look as if he's had a heart attack. It ...
This is almost certainly The Greatship, a short story collection (and likely the story titled "Alone") by Robert Reed
There is a shape-shifting robot called Alone, and it does eventually work its way into the ship to explore, while hiding from the inhabitants.
The 4th story is "Born with the Dead", a 1974 novella by Robert Silverberg, which was the (unaccepted) answer to this old question. It was first published in The Magazine of Fantasy and Science Fiction, April 1974, which is available at the Internet Archive. Here's a description from Majipoor.com:
In the 1990s, doctors have discovered how to rekindle dead ...
I know this is a very old post, but I believe this might be "A Taste for Dostoevsky" by Brian Aldiss.
Looking up, he saw that a solitary figure stood on a ridge of rock, staring moodily up at the fake heavens. He identified it as Cat Vindaloo, the Pakistani director of their show, and called a greeting to him.
Cat nodded sourly and altered his ...
Probably The Empire of T'ang Lang, Alan Dean Foster, in the anthology With Friends Like These...
Similar title, protagonist is a mantis.
The story is also found in the anthology The Alien Condition, and this book has a cover like you describe:
This one would be difficult to find using Google. Most searches for Foster and insects will return nothing but ...
Franz Rottensteiner's View From Another Shore contains a story entitled "A Modest Genius" by Vadim Shefner , which does cause the whole world to grow dark. It is a collection of short stories which was published in 1973, which matches your timeframe.
Here's the opening to that story:
Sergei Kladesev was born on Vasilyevski Island, Leningrad. He was a ...
This is "Stranded" (2012) by Anne Bishop.
The Restorers travel the universe fulfilling a purpose handed down
through the generations. They live and die aboard city-ships, never
knowing the worlds they create and save. What begins as a disastrous
training exercise in creating and balancing ecosystems becomes an
unexpected fight for survival. The ...
"Nerves" and "Who Goes There?" both appear in the famous anthologies "Adventures in Time and Space" (1946) (aka "Famous Science-Fiction Stories") and "The Science Fiction Hall of Fame Volume Two" (1973).
Neither of them, however, contains Sheckley's "Specialist" as well.
Piers Anthony's novel Prostho Plus is a fix-up of his series of short stories about a prosthodontist named Dr. Dillingham who is abducted by aliens and forced to travel around the universe fixing the teeth of various alien monsters.
One of those short stories, "In the Jaws of Danger", appeared in an anthology called Young Extraterrestrials which featured a ...
The first short story is "Cocoon" by Keith Laumer, which was also the answer to this question. It was first published in Fantastic Stories of Imagination, December 1962, available at the Internet Archive.
An excerpt from "Cocoon":
A face appeared. This was a different one, Sid was sure. It was hairier than the other one, but not as hollow-cheeked. He ...
Andromeda I (1976), edited by Peter Weston
The title doesn't match but both stories you mention appear to be inside of it.
"Doll" by Terry Greenhough
Next up is Doll by Terry Greenhough. What can I say about this one? Well, if you are up for an extended description of an alien birthing ceremony involving ‘dolls’ that Cyric the moulder shapes to ...
The first story sounds like "The Edge of the World" by Michael Swanwick, previously identified as the answer to this question and this one. Quoting from the description in Daphne B's question:
In this story, three kids are going through a wild area, I believe mountainous, in order to find a thing/being that grants wishes. There are two boys and a girl. ...
One of the stories might be 'A Walk in the Dark' by Arthur C. Clarke. You could look for anthologies containing it.
Remember that feeling of turning out the lights and walking up the stairs? Remember the moment when childhood bravado fails and the what if's begin - what if there are monsters in the dark, what if the darkness makes them bold, what if they ...
The first story, as suggested in a comment by Mr Lister, is probably "Pawley's Peepholes" (aka "Operation Peep"), a short story by John Wyndham published in Science-Fantasy, Winter 1951, which is available at the Internet Archive. Set in the present (i.e. the middle of the 20th century, when it was written), it is about annoying time-tourists from the future....
Otherwere: Stories of Transformation
These masterful tales do not involve werewolves, butr a wonderfully diverse assortment of creatures including a serpent, a child, an elephant--even a right-wing Republican. Blending elements of fantasy, mystery, comedy and mythology, these extraordinary tales take lycanthropy into an entirely new realm.
The elephant ...
I have managed to track down the book.
It is "Beyond Infinity" by Robert Spencer Carr.
Turns out that the story by Arthur C Clarke was not even in it... My memories are all muddled the older I get...!
Appreciate all your help.
The story would appear to be "The Master" (1989) by Diana Wynne Jones. It was originally published in Hidden Turnings but considering you think you read it in an anthology of stories by the same author I imagine you actually read it in Unexpected Magic: Collected Stories. The reviews on Goodreads mention brief overviews of the story but I have found the ...
A WorldCat search for a book with both Jansson and Tolkien in the author credits identifies Children's classics to read aloud, published in 1992.
The contents include; Under the hill (from The Hobbit by J.R.R. Tolkien) and
Invisible Child (from Tales from the Moomin Valley by Tove Jansson).
This could be The Golden Treasury of Children's Literature, first published in 1947 by Bryna and Louis Untermeyer, with various editions and volumes published since then.
According to this link, volume 5 "Wonder Lands" contains an excerpt from "The Hobbit" and "The Happy Moomins" by Jansson.
The "hollowed out asteroid" story is Stableford's "The Engineer and the Executioner". It was asked about in this question: Trying to remember the name of a short scifi story about a biology experiment in space but it is not a duplicate, because this question is asking about an anthology.
Based on that story, I think the anthology may be "Sexual Chemistry: ...
This is only a partial answer I'm afraid, but the escalator story sounds like Descending by Thomas M. Disch. I first read it in a 1977 anthology called Decade the 1960s, but there is no story by Borges there and I can't see one about a man losing his head. I have Googled and Googled, but I cannot find any anthology containing both stories.
Here is a long list of places where "Nerves" was published, although the only possibilities are those published in or before the 1970s.
Here is a long list of places where "Who Goes There?" was published, although the only possibilities are those published in or before the 1970s.
Just a guess, but there's a story "Three in a Bed" in the anthology Echoes of Terror, which is at least kinda purplish.
It also appears in The Ghost Book of Charles Lindley and Lord Halifax's Complete Ghost Book.
It looks like David Drake addressed this on his website:
For a setting I used a young nobleman making a grand tour with his tutor and personal servant. The action stemmed from real events in the Greek islands in 1795: a Russian nobleman determined to take a Greek temple back to St. Petersburg in the warship he borrowed from the Czar–and who bombarded it ...
Damon Knight edited an anthology titled First Contact which was published in 1971 by Pinnacle Books -- it was reprinted at least twice: ISFDB.org. Might this be it? (There were other anthologies with this title, but this is by far the most famous and IMO the best.)
Contents (from isfdb):
7 • Introduction (First Contact) • (1971) • essay by Damon Knight
A Google search on fairies bluebell sapphire goblin maid short stories turns up the answer: Shadow Castle by Marian Cockrell.
Quoting the Amazon description:
In the middle of a deep forest is an enchanted valley and a castle where only shadows live, shadows of kings and queens who have waited for hundreds of years for the spell cast upon them to be ...
The Beastly Bride - Tales of the Animal People (2009)?
This review matches a number of the stories you mentioned.
A fire salamander girl getting stolen and taken to hell, where her creator had to rescue her.
Marly Youmans’ “The Salamander Fire” takes us through farmer’s markets, the mysteries of glassblowing, love and hell in an odd blend of the myths ...