I believe the book you're looking for is Possible Worlds of Science Fiction, edited by Groff Conklin. This anthology is divided into two parts: Part One with 12 solar system stories (one story each for the nine planets, the sun, the moon, and the asteroid belt), Part Two with 10 interstellar stories. There are other anthologies with a story for each planet of the solar system, but this seems the only one that includes the asteroid story you described, which is positively identified below.
The Mercury story in this book is "Cones" by Frank Belknap Long; first published in Astounding Stories, February 1936, available at the Internet Archive. It does not exactly fit your description, as there are no sentient pools of molten metal. Quoting from Everett F. Bleiler's review of "Cones" in Science-Fiction: The Gernsback Years:
The greatest dangers in exploring seemingly lifeless Mercury are hot spots, areas where ultraviolet radiation suddenly bursts forth with disintegrating force. Such hot spots kill several members of the expedition. A second problem, now first discovered, consists of tall, cone-shaped semi-intelligent creatures that move so rapidly that the eye cannot catch them, though Crayley photographs them with a stroboscopic camera. Hostile, they project a beam of ultraviolet light that dissolves the calcium in the human body. These beings account for the rest of the expedition except for Gibbs and Mona, who are lucky to escape.
The asteroid story is "Asleep in Armageddon" aka "Perchance to Dream" by Ray Bradbury; first published in Planet Stories, Winter 1948, available at the Internet Archive; reprinted (among other places) in Avon Fantasy Reader, No. 11, 1949, also available at the Internet Archive. It was also the subject of this question.
A spaceman is stranded on an asteroid:
Sale pried himself out of the small airlock and stood breathing the atmosphere. Luck. Sheer luck. The air was breathable. An instant's checking showed him that he had two months' supply of food with him. Fine, fine! And this—he fingered at the wreckage. Miracle of miracles! The radio was intact.
He stuttered out the message on the sending key. CRASHED ON PLANETOID 787. SALE. SEND HELP. SALE. SEND HELP.
The reply came instantly: HELLO, SALE. THIS IS ADDAMS IN MARSPORT. SENDING RESCUE SHIP LOGARITHM. WILL ARRIVE PLANETOID 787 IN SIX DAYS. HANG ON.
Warring ghosts invade his dreams:
At last, at last, sang a high voice. Now, now. The long time, the waiting. Over, over, sang the high voice. Over, over at last!
It was like being under sea. Green songs, green visions, green time. Bubbled voices drowning in deep liquors of sea tide. Far away choruses, chanting senseless rhymes. Leonard Sale stirred in agony.
Mine, mine, cried a loud voice. Mine, mine! shrieked another. Ours, ours! shrieked the chorus.
The din of metal, the crash of steel, the conflict, the battle, the fight, the war. All of it exploding, his mind fiercely torn apart!
He leaped up, screaming. The landscape melted and flowed.
A voice said, "I am Tylle of Rathalar. Proud Tylle, Tylle of the Blood Mound and the Death Drum. Tylle of Rathalar, Killer of Men!"
Another spoke, "I am Iorr of Wendillo, Wise Iorr, Destroyer of Infidels!"
The chorus chanted, *And we the warriors, we the steel, we the warriors, we the red blood rushing, the red blood falling, the red blood steaming in the sun—
Leonard Sale staggered under the burden. "Go away!" he cried. "Leave me, in God's name, leave me!"
Eeeeeeeee, shrieked the high sound of steel hot on steel.
He stood with the sweat boiling out of him. He was trembling so violently he could not stand. Insane, he thought. Absolutely insane. Raving insane. Insane.
He tries to stay awake:
Now that he was feeling better, it was all silly. All I have to do, he thought, is stay awake six nights. They won't bother me that way. When I'm awake, I'm dominant. I'm stronger than those crazy monarchs and their silly tribes of sword flingers and shield bearers and horn blowers. I'll stay awake.
But can you? he wondered. Six whole nights? Awake?
There's coffee and medicine and books and cards.
But I'm tired now, so tired, he thought. Can I hold out?
Well, if not . . . There's always the gun.
P.S. M.A. Golding's answer suggests that your Mercury story may be Robert Silverberg's "Sunrise on Mercury", which first appeared in Science Fiction Stories, May 1957 (available at the Internet Archive) and in New Worlds Science Fiction #60, June 1957 (also available at the Internet Archive). This seems likely. As far as the ISFDB knows, there is no compilation containing both Silverberg's "Sunrise on Mercury" and Bradbury's "Perchance to Dream" = "Asleep in Armageddon". However, Silverberg's story definitely features a sentient pool of molten metal:
It was a great pool of molten zinc, lying shimmering between two jagged crests somewhere on Sunside. It had been there thousands of years; it would be there thousands, perhaps millions of years from now.
Its surface quivered. The sun's brightness upon the pool was intolerable even to the mind's eye.
Radiation beat down on the zinc pool—the sun's radiation, hard and unending, and then a new radiation, an electromagnetic emanation and with it a meaningful commutation:
I want to die.
The pool of zinc stirred fretfully with sudden impulses of helpfulness.
The vision passed as quickly as it came. Stunned, Ross looked up hesitantly. The expression on the six faces surrounding him told him what he wanted to know.
"You felt it too," he said.
Spangler nodded, then Krinsky and the rest of them.
Brainerd turned to Spangler. "Are we all nuts, Doc?"
The psych officer shrugged. "Mass hallucination . . . collective hypnosis . . ."
"No, Doc." Ross leaned forward. "You know it as well as I do. That thing was real; it's down there, out on Sunside."
"What do you mean?"
"I mean that wasn't any hallucination we had. That's life—or as close to it as Mercury can come." Ross' hands shook; he forced them to subside. We've stumbled onto something very big," he said.
Spangler stirred uneasily. "Harry—"
"No, I'm not out of my head! Don't you see—that thing down there, whatever it is, is sensitive to our thoughts! It picked up Curtis' godawful caterwauling the way a radar set grabs electromagnetic waves. His were the strongest thoughts coming through; so it acted on them and did its damnedest to help Curtis' wish come true."