The Byzantine empire had been on the decline at this point for 200 years. Some would say that it ended after the 4th crusade when Constantinople was sacked by the crusaders. The Palaiologos family were the last dynasty of a crumbling empire. The Ottoman Turks conquered the Anatolian heartland and parts of the Balkans before finally marching on Constantinople. The last emperor, Constantine XI Palaiologos, appealed to the Christian kingdoms in the west for help, but received little to none. His coffers were depleted so he couldn't raise mercenary armies as in days of old. When the Ottomans began assembling outside the walls of the city, the Byzantines knew only an act of G-d would help them, but they received none. As long as the Theodosian walls held tho, and the golden horn wasn't breached, they could hold it the city for a long time. But they were outmatched and outgunned, and eventually the walls were breached and the Byzantine empire was ended.

A great book on this topic is 1453 by Roger Crowley, which talks about the months leading up to May 29, and will answer the rest of your questions.