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So I'm a little confused as to the Klingons and how their lifespan and stages are.

According to Memory Alpha:

"Klingon children matured far more quickly than Human children. At the age of only one Earth year, a Klingon child had the appearance a Human child had at about four. By the age of eight Earth years, a Klingon attained the maturity a Human did not reach until about age sixteen. (TNG: "Reunion"; DS9: "Sons and Daughters") Rop'ngor was a common childhood disease among Klingons. (TNG: "Up The Long Ladder") When Klingon children began growing into adults, they went through jak'tahla, a Klingon form of puberty. (Star Trek: Insurrection) Like other mammalian species, Klingon females were capable of lactating to breast-feed infants. (TNG: "A Matter Of Honor")"


So I guess Klingons start military training literally at the age of 8 Earth years or something like that?

Basically, they're trying to explain Alexander's growth spurts. He starts out very young looking when his mother dies. Then turns into an older boy for a while and then comes back in DS9 looking very grown up.