This looks like a fun project.
Earlier in this thread I posted a complete list of REH stories in WT including the Conan stories so that covers that.
Skull Face and Others from 1945 does have a few Conan stories - "The Hyborian Age," "The Phoenix on the Sword," "The Scarlet Citadel," "The Tower of the Elephant," "Rogues in the House," and "Shadows in Zamboula." As far as know, this is the first time any of the Conan stories were published in book form.
Avon Fantasy Reader published several REH stories in the late forties and early fifties including two Conan stories, both in 1949 - "Queen of the Black Coast" (no. 8) and "A Witch Shall be Born" (no. 10).
Ace published "Hour of the Dragon" in 1950 under the title Conan the Conqueror as a double paperback (the other side is The Sword of Rhiannon by Leigh Brackett).
Next came the Conan series published by Gnome (see Tupenny's sweet collection above for a visual reference). They were:
Conan the Conqueror (1950)
The Sword of Conan (1952)
The Coming of Conan (1953)
King Conan (1953)
Conan the Barbarian (1954)
Tales of Conan (1955)
The Return of Conan (1957)
These contained the Conan stories that were edited by L. Sprague de Camp and Lin Carter that were the basis for the later Lancer/Ace paperback series that most of us probably grew up with. Tales of Conan is all non-Conan stories by REH that were rewritten by de Camp as Conan stories. The Return of Conan is not by REH but is the first Conan pastiche by Bjorn Nyborg.
Several Conan Stories were published posthumously for the first time (all edited by de Camp):
"The God in the Bowl" in Space Science Fiction, Sept. 1952.
"The Frost-Giant's Daughter" in Fantasy Fiction, Aug. 1953.
"The Treasure of Tranicos (aka The Black Stranger)" in Fantasy Magazine,
Mar. 1953.
"The Vale of Lost Women" in The Magazine of Horror, Spring 1967.
Next would be the Lancer/Ace paperbacks and the Donald Grant editions in the 60s and 70s, but I need to get to bed so those will have to wait for another time.