I believe you're looking for Galactic Empires Volume One (1976), edited by Brian W. Aldiss. (Volumes One and Two were combined for the 1988 Galactic Empires, which is another possibility if you don't recognize the cover of Volume One.)
The first story is "The Rebel of Valkyr" (1950) by Alfred Coppel.
The hero is Kieron, Warlord of Valkyr, who is turned away by the Emperor's widow, the Consort Ivane "the Fair," (who Kieron refers to as "Ivane the schemer") when he tries to petition for aid for his people. Leaving the palace he runs into Alys, daughter of the Emperor, who he last saw as a girl but who is now a young woman, who is being kept incommunicado in the palace.
Kieron goes to meet with Freka, Lord of Kalgan (Freka the Unknown) who is convening a grievance council of star-kings. Freka tries to lull Kieron and the other star-kings of the Outer Marches with wine, women, food and entertainments, but Keiron observes that he's underestimating the fortitude of "the men of the Edge." Kieron finds Freka inhumanly perfect but also somehow lacking; Freka nevertheless stirs the star-kings, except Kieron, to rebellion.
Alys shows up, having been saved from an attack made by soldiers of Kalgan, and Kieron discovers Ivane's lackey Landon is sneaking around. Kieron deduces the rebellion is a scheme of Ivane's to create justification to take the Imperial throne herself. He and his guard sacrifice themselves to let Alys escape, and Kieron is captured and imprisoned.
Alys and his soldiers return and frees him, and they head to Earth where Kieron confronts Freka, demonstrating that he is an android. Ivane kills herself rather than be taken prisoner, Alys becomes the Empress and Kieron leaves for home.
The second story mentioned is "The Crime and the Glory of Commander Suzdal" (1964) by Cordwainer Smith.
Commander Suzdal is an explorer, sent out in a shell-ship for a thousands-year-long flight. He spends almost all the trip in frozen sleep while manufactured beings ("turtle men") live generations maintaining the ship. He is wakened only for emergencies, one of which is the discovery of a distress capsule from Arachosia. Which is a trap that he unthinkingly falls into when he follows it back to its origin, and is captured along with his ship by the Arachosians, who wish to destroy all humanity.
With his ship immobilized, his weapons blocked and his self-destruct not certain enough to prevent the Arachosians from finding Earth, Suzdal takes cat embryos and encodes them with directives to make them into a hostile intelligent race - and then throws them two million years into the past.
For his crimes Suzdal is stripped of name, rank, everything except his shame, and sent into oblivion.
The final story you recall is "All the Way Back" (1952) by Michael Shaara.
A human ship with two crew is exploring for planets, finding none after years of searching. An alien ship, observing the human ship, wonders at their appearance in "Great Desert of the Rim." The star Mina has seven planets, of which the second is habitable, and the human ship lands there. The alien ship, a patrol ship of the Galactic Federation, follows them and the galactics confront the humans.
The galactics tell the humans the story of the race called the Antha, who arose in the region of space known as the Desert, and who almost conquered the entire galaxy before they were finally suppressed, their planets destroyed, their stars turned into novae. One of the notable acts of cruelty performed by the Antha was to genetically engineer the entire Apecta race to be missing an arm.
The patrol captain, Roymer, tells the humans that the Antha have now reappeared before he kills them.
All three stories can be read in the Internet Archive copies of their original magazine printings:
