"The girl designs shoes and socks."
Translation:Pigen designer sko og strømper.
No, sko doesn't change from the indefinite singular to the indefinite plural.
I don't think there's a specific reason. English has words like this too, such as sheep, deer and aircraft.
And no, sko isn't a loan word. It developed from the same root as English shoe, German Schuhe and Dutch schoen.
I wonder if "strømper" might be related to the archaic English "strumpet" (whore)? The image of a woman rolling up her stockings is titillating even to modern audiences; that could explain the connection, if there is one.