He doesn't bother himself (about) cooking [to cook].
彼はわざわざ自分で料理などやらない(◆oneselfを略すことも多い)
He doesn't drink very much.
彼はあまり飲むほうではない(◆「大量に」または「しばしば」の意)
He doesn't fit the romanticized role of a cowboy.
彼に美化されたカウボーイ役は合わない
He doesn't give [care] a rap about that.
彼はそんなことは気にもしない
He doesn't like beef, and no more do I.
彼は牛肉が嫌いだが,私も嫌いだ(... and neither do I.)