The victory of the minamoto marks the beginning of what period in japanese history?

Victory over the minamoto marked the beginning of Feudalism. This is a dominant social system used in medieval Europe where nobility held ownership of the lands in exchange for military service from the people. These people were eventually turned into the tenants of the nobles and the serfs were allowed and mandated to live on the lands of the lords and nobles in exchange for labor, a share of their crops or other produce, military protection and homage.

It is accepted that the Byzantine Empire fell in 1453 to the Ottomans under Mehmed II finally took over Constantinople after 53 days of battle.  However, there were several other successor states to the Byzantine Empire such as the Despotate of Morea and the Empire of Trebizond that could be argued to be "Byzantine".  Some didn't even fall until 1479. That being said, the Roman "Byzantine" Empire definitely fell in 1453.