Short answers, pulled from the story.
Constantine XI Palaiologos was born on the 8th of February 1404 to Emperor Manuel II Palaiologos and Helena Dragaš. He was born inside the marble halls of Constantinople within the imperial palace.
John VIII Palaiologos led a campaign against Carlo I Tocco in 1427 bringing his brother Constantine along as Despot of the Morea. The brothers captured Glarentza from the Epirotes and defeated Tocco at the Battle of the Echinades off the coast of Glarentza.
Constantine died fighting in battle on the following day after Constantinople fell on the 29th of May 1453. Most historical agreements state the emperor perished leading the last stand against Ottomans.
Pope Nicholas V viewed Church Union as far more pressing than Ottoman threats while most Western kingdoms remained occupied with their own conflicts. Venetians threatened to transfer trade operations elsewhere after Constantine raised taxes on goods they imported into Constantinople.
Work commenced on building Rumelihisarı castle positioned on western Bosporus strait opposite existing Anadoluhisarı fortress in spring 1452. With both castles controlling sea traffic in the Bosporus Mehmed could blockade Constantinople simultaneously by land and sea.