Is alcoholism recognized by the APA as a mental illness?

Question

but is it a mental illness alone, or only when combined with another mental illness?

Answer 1

yes, there are two diagnostic categories that cover it-alcohol dependence and alcohol abuse-depending on the level of severity and what criteria you meet.
Alcohol dependence specifies a withdrawal syndrome and development of tolerance (indications of which are needing to take increasing amounts in order to achieve the same effect). Alcohol abuse specifies negative results arising from use (DWI's, fights, contraindications with meds, etc.) and continued use despite negative consequences.

Answer 2

yes