Short answers, pulled from the story.
Alexios Komnenos and his brother David founded the Empire of Trebizond in April 1204. They seized the city of Trebizond with troops provided by their aunt Queen Tamar of Georgia just weeks before the Fourth Crusade sacked Constantinople on the 13th of April 1204.
The Empire of Trebizond stretched along the southern coast of the Black Sea and into the western half of the Pontic Mountains. Its territory included Chaldia and extended eastward to Akampsis, covering modern Turkish provinces including Sinop, Samsun, Ordu, Giresun, Trabzon, Bayburt, Gümüşhane, Rize, and coastal parts of Artvin.
Rulers employed marriage alliances as their primary diplomatic tool by offering daughters famed for their beauty to Turkish dynasts with generous dowries. Alexios IV married two daughters to Muslim rulers Jihan Shah khan of Kara Koyunlu and Ali Beg khan of Ak Koyunlu while another daughter became the third wife of Byzantine Emperor John VIII Palaiologos.
Mehmed II conquered the city of Trebizond on the 15th of August 1461 after a one-month siege. This event ended the last independent remnant of the Byzantine Empire and the Roman Empire from which it sprang.
Trebizond functioned as a vital entrepôt on the southeastern terminus of the historic Silk Road and accumulated wealth under Mongol suzerainty due to its link with Tabriz. The empire exported Pontic silver alum wine and hazelnuts while issuing silver coins known as aspers between 1238 and 1263.