Short answers, pulled from the story.
Andreas Palaiologos was the eldest son of Thomas Palaiologos, Despot of the Morea, and the nephew of Constantine XI, the final Byzantine emperor. After the fall of Constantinople in 1453 and his father's death in 1465, he became the senior male heir of the Palaiologos dynasty and the principal claimant to the Byzantine imperial title. From 1483 onwards he styled himself Emperor of Constantinople.
Andreas Palaiologos died in Rome in June 1502. He was buried with honor in St. Peter's Basilica, next to his father Thomas. His widow Caterina received 104 ducats from Pope Alexander VI to cover the funeral expenses.
In 1494, Andreas sold his claims to the thrones of Constantinople and Trebizond to Charles VIII of France. The transfer was arranged by French Cardinal Raymond Peraudi and formalized on the 6th of November 1494 at the Church of San Pietro in Montorio in Rome. When Charles died in 1498 without fulfilling the agreed conditions, Andreas reclaimed the titles.
The main cause was systematic reductions to his papal pension over decades. His father Thomas had received 300 ducats a month plus 200 from the cardinals; by 1492, Andreas's pension had been cut to just 50 ducats a month under Pope Alexander VI. Historian Jonathan Harris argued in 1995 that this progressive reduction, not personal extravagance, was the primary reason for Andreas's poverty.
The expedition was organized in southern Italy following the death of Sultan Mehmed II and the civil war between his sons Cem and Bayezid. Andreas assembled men at Foggia, hired mercenaries including the Greek soldier Krokodeilos Kladas, and received a letter of support from Pope Sixtus IV on the 15th of September 1481. The expedition was abandoned because Bayezid stabilized his rule, the major Christian kingdoms were disunited, and Venice had recently signed a peace treaty with the Ottomans.
No confirmed descendants are known. A 1487 Roman epitaph mentions a Lucretia described as his daughter, but she is recorded as dying at 49 years of age, which is impossible given Andreas's own birth date. Possible candidates include a son called Fernando mentioned by Ludovico Sforza in 1499 and a son called Constantine recorded in the Papal Guard, but neither claim is conclusively established.