The decapitated head hanging by his own hair in the darkness is from 1935's "The Servants of Bit-Yakin" (frequently anthologized under the alternative title "Jewels of Gwahlur"), another one of the original Conan stories by Robert E. Howard:
As noted in the other answer, the scene with the woman who betrays Conan thrown into the cesspool is from another Howard story story, "Rogues in the House" (1934).
The wench was sitting cross-legged in her shift on her unkempt bed. She turned white and stared at him as if at a ghost. She had heard the cry from the stairs, and she saw the red stain on the poniard in his hand. But she was too filled with terror on her own account to waste any time lamenting the evident fate of her lover. She began to beg for her life, almost incoherent with terror. Conan did not reply; he merely stood and glared at her with his burning eyes, testing the edge of his poniard with a callused thumb.
At last he crossed the chamber, while she cowered back against the wall, sobbing frantic pleas for mercy. Grasping her yellow locks with no gentle hand, he dragged her off the bed. Thrusting his blade in the sheath, he tucked his squirming captive under his left arm and strode to the window. As in most houses of that type, a ledge encircled each story, caused by the continuance of the window ledges. Conan kicked the window open and stepped out on that narrow band. If any had been near or awake, they would have witnessed the bizarre sight of a man moving carefully along the ledge, carrying a kicking, half-naked wench under his arm. They would have been no more puzzled than the girl.
Reaching the spot he sought, Conan halted, gripping the wall with his free hand. Inside the building rose a sudden clamor, showing that the body had at last been discovered. His captive whimpered and twisted, renewing her importunities. Conan glanced down into the muck and slime of the alleys below; he listened briefly to the clamor inside and the pleas of the wench; then he dropped her with great accuracy into a cesspool.