In August 1862, the age limits under the Conscription Act were 18 and 35.
The term "non-conscript" applied only to the provisions of the Conscription Act. Discharges for wounds, injuries or illness were based on a surgeon's certificate of disability. This was a statement by an army surgeon which included a diagnosis of the injury or illness, a prognosis as to the estimated length of time it would take the soldier to recover (if ever), and a recommendation for a disability discharge or a convalescent leave (whichever applied).
Disabled Confederate soldiers were pretty much on their own after being discharged, and had to try to make a living as best they could within the limits of their disability. Later in the war, however, the War Department instituted the "Invalid Corps." This allowed disabled soldiers to remain in the service with pay and allowances to perform non-combat duties. For example, Capt. George Douglass Alexander, of the 3rd Arkansas Regiment, who lost his left arm in a skirmish in Virginia, was retired to the Invalid Corps, where he was promoted to major and served as ordnance-master at Arkadelphia and Marshall, Texas, for the rest of the war.
Of course, there were a few disabled soldiers who were able to convince the army surgeons that they were able for active field service, despite their disability. The most notable example, I guess, was Lieutenant-General John Bell Hood, an amputee, who remained in the service and was able to campaign by having an aide strap him to the saddle.