What Kage did Itachi kill

Why did the Uchiha clan want to rebel?

Why did [the Uchiha clan] want to rebel?

The Uchiha clan rebelled over the many adverse conditions imposed by the Konoha village leadership on their clan and contributed to the mutual distrust between the two factions.

The Konoha leadership, starting with Tobirama, the second Hokage, had mistrusted the Uchiha clan, and over time they mistrusted each other more and more.

According to Itachi Shinden, no one from the Uchiha clan was named a Hokage after the third Shinobi World War, despite their contributions to the war. And after the attack by the Nine Tails on Konoha, it was suggested that they were indeed the cause (since the Sharingan is known to be able to control the Nine Tails like Madara).

After the Nine Tails attack, the Konoha Police Headquarters (which was badly damaged) and the Uchiha clan's houses were moved to the edge of the village, too far from the center of the village, despite Fugaku Uchiha's protests. (Fugaku is the head of the Uchiha clan and the father of Itachi and Sasuke.) The place can also be easily spied on from multiple angles, adding to the distrust between the Uchiha clan and the Konoha council and leading to the massacre of the Uchiha- Clan leads.

The third Hokage doesn't seem to be the kind that condones genocide. Does this make Konoha a socialist country?

I'm not sure why you linked socialism with the third Hokage, which condones or does not condone genocide. About the third Hokage it was not his order to murder almost the entire Uchiha clan, but Danzo Shimuras, one of the elders of Konoha. The third Hokage advocated a diplomatic solution to the situation.

Konoha is part of the land of fire (a land) ruled by a daimyo. This is similar to the form of government in feudal Japan.