Hi, Rebecca.
Partisan Rangers were units authorized by the Confederate Congress in April 1862. They were intended to be recruited locally in Federal-occupied areas, for service in the area where they were recruited. A behind-the lines sort of unit. Most were ineffective, and poorly disciplined. General Lee urged that all except John Mosby's (43rd Virginia Cavalry Battalion) and Hanse MacNeill's be disbanded, and the men brought into regular Confederate units, and this was done, by and large.
I have never heard of mountain rangers.
The militia were units formed during peacetime. In theory every white man between 18 and 35 was in the militia. Militia units formed up in 1861, were formed into regiments (10 companies per regiment) and accepted into Confederate service. In April 1862, the Confederate Congress declared universal white male conscription, so most white men had to join up, or be drafted. After this, the militia still existed, but was composed of men who had been awarded exemptions, or those over 35 (later 45) and under 18 (later 17).
In terms of organization, they were generally the same as volunteer regiments (i.e. ten companies), but they were only called out during a particular crisis. Stonewall Jackson, for example, called out the militia in the Valley companies in the spring of 1862.
The militia mostly consisted of infantry units (because they were cheaper and the drill was less demanding), although there were some militia artillery, and may have been some cavalry units. Maybe somebody else can enlighten us.

As for "Ashby's Cavalry," in Confederate service, units retained the name of a original or popular commander, even after that commander's promotion to command of a large unit, or his death. The 7th Virginia Cavalry, however, truly was Ashby's, because it took on his personality, and retained it after his death.
I hope this helps.
Jonathan White