Usually, ant names are base on words from another language, usually Greek or Latin, but often others, and the names are intended have a meaning. Myrmica, -idris, -myrmex, etc are various ways of saying ant in Greek, while Formica means ant in Latin. Species names such as rubra, longispinosus, etc. describe characteristics, in these cases red color and long spines. The has been an unfortunate trend recently by some ant taxonomists to coin names that are simply combinations of of letters that look like Greek or Latin but are actually nonsense words. I find these harder to remember than names with meaning, but I suppose for most people, who do't know any Latin or Greek it doesn't matter much.
The endings "rmex,cine,mica,tus,sius" you cite do not have meaning themselves. They are mostly grammatical endings, but in these you have also "stolen" a letter form the root word and put it on the front end of the grammatical ending, like asking what do the -ding of ending, or the -len of stolen mean?