This is the 1962 Ace paperback edition of a 1957 hardcover anthology, The Best from Fantasy and Science Fiction, Sixth Series, edited by Anthony Boucher. The cover by Ed Valigursky is inspired by Poul Anderson's story "The Man Who Came Early". Note the letters MP on the soldier's helmet.
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 ...
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 ...
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 ...
"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.
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 ...
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 ...
The first story is probably "Housing Problem" by Henry Kuttner and C.L. Moore, and the third sounds like "The Truth About Pyecraft" by H. G. Wells. (I recognized both from having read them.) An older "big hardback" that contains both of those stories is ALFRED HITCHCOCK'S GHOSTLY GALLERY from 1962:
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 ...
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 ...
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 ...
"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.
A complete collection of all of Asimov's fiction was planned, but abandoned after only two volumes.
According to Isaac Asimov in the introduction to The Complete Stories, Volume 1, Doubleday intended to publish a uniform collection of all of his fiction: short stories and novels, science fiction and mysteries:
It is time, therefore, for Doubleday to pull ...
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 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. ...
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 ...
As per Short story about a mutiny on a spaceship and a polite captain, this is probably "Mutiny" by Malcolm Hulke, which only appeared in one anthology, Purnell's Book of Adventure in Space, also released as Galactic Adventures.
click image to embiggen
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 ...
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 ...
Dracula: Prince of Darkness
This anthology of vampire tales was published in 1992, so it fits the time frame. The story you remember is probably “The Lord’s Work,” which features a vampire-killing nun. It seems to match the details you remember:
The nun does say “I’ll stop when they stop.” She also has an internal voice like the one mentioned in the linked ...
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.
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.
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.
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.
The first story is The First Men (1960) by Howard Fast.
Fast's story is about how a group of scientists and educators, through a controlled environment, succeed in raising naturally gifted children into "man-plus"—people who possess comparatively super-human abilities. They possess unparalleled understanding of all technical subjects such as math, physics ...