To answer your questions, the Confederate Army of the Mississippi was renamed the Army of Tennessee in November of 1862.
Lead elements of the Confederate army under Kirby Smith entered Kentucky during late August of 1862. Bragg's army followed a couple of weeks later, other parts of Kirby Smith's command laying siege to Cumberland Gap. Stevenson's Division fought at Tazewell TN and participated in the investment of Cumberland Gap. Bragg's army captured Munfordville and moved on to Bardstown while Kirby Smith occupied Lexington and Frankfort. After the Federal command at Cumberland Gap relinquished that fortress, Stevenson's men marched into Kentucky.
Part of the Confederate failure in Kentucky may be attributed to a divided command structure. It's easy to assume a single command structure because it's the obvious choice for such an invasion. Unfortunately Bragg fought at Perryville while Kirby Smith was still in the Frankfort-Lexington area. By the time both armies united, Bragg had made a decision to withdraw.
As a result of this lost opportunity, the War Department sent General Joseph E. Johnston to direct and coordinate activities of Confederate armies in Mississippi, Alabama, Tennessee and north Georgia. Thomas Connelly's Army of the Heartland: The Army of Tennessee, 1861-1862 will provide further explanation of these and other Confederate woes from that period.