Short answers, pulled from the story.
The Latin Empire replaced the Byzantine Empire in April 1204 when the crusader army sacked Constantinople. This event marked the end of the original plan to restore Isaac II Angelos to his throne after he was usurped by Alexios III Angelos.
Baldwin IX of Flanders became the first emperor of the Latin Empire selected from among the crusader ranks. He placed Rom on his seals leaving open whether he meant land of the Romans or the Romans themselves while later rulers called the empire Emperor of Romania.
Emperor Baldwin was imprisoned in Tarnovo until his death later that year after Kaloyan crushed the Latin heavy cavalry with Cuman allies at the Battle of Adrianople. The conflict occurred on the 14th of April 1205 and resulted in the loss of significant territory for the new state.
The Latins dismantled the Greek economic administration immediately after conquest disrupting all forms of production within the city. By the 1230s Constantinople faced a major shortage of basic foodstuffs despite its reduced population and relied on selling looted religious relics to Western Europe as its only significant export basis.
Alexios Strategopoulos recaptured Constantinople on the night of the 24th of July 1261 when he led an advance force of 800 soldiers mostly Cumans into the city. A detachment entered through a secret passage near the Gate of the Spring giving Nicaean forces entry while the Latins were taken completely unaware.