Short answers, pulled from the story.
The Trans-Mississippi Department was a territorial command of the Confederate States Army covering Arkansas, Louisiana west of the Mississippi River, Texas (including present-day New Mexico and Arizona), and the Indian Territory. It was established on the 26th of May 1862 under War Department General Orders No. 39.
General E. Kirby Smith was the longest-serving and most consequential commander, holding the post from the 7th of March 1863 through the department's final surrender. Before him, Brigadier-General Paul O. Hebert, Major-General Thomas C. Hindman, and Lieutenant-General Theophilus H. Holmes each held the command briefly.
After Union forces captured Vicksburg and Port Hudson and closed the Mississippi River to Confederate traffic, General E. Kirby Smith was cut off from the Confederate capital at Richmond. He was forced to run the department as a nearly independent command, negotiating directly with foreign countries. Confederates used the nickname "Kirby Smithdom" to reflect how completely he governed the region.
The Trans-Mississippi Department surrendered on the 26th of May 1865, making it the last Confederate department to yield to United States forces at the end of the American Civil War.
The department opened with headquarters at Little Rock, Arkansas. On the 24th of April 1863, it moved to Shreveport, Louisiana. On the 18th of May 1865, it relocated a final time to Houston, Texas, where it remained until the surrender.
The department included Arkansas, the portion of Louisiana west of the Mississippi River, Texas (encompassing lands that are now New Mexico and Arizona), and the Indian Territory.