Is this word ever shortened to "el lab" as in English?
I don't know. I've never heard anyone say el lab.
I doubt it. They tend to talk with their words ending in "o" or "a". I had wondered the same thing with "televisión".
La televisión is shortened to "La tele".
Sometimes the article can be omitted when translating and other times not. Why was just "laboratory" wrong this time?
if you see the spanish has the article and the noun, but the noun is not a subject (its an object), then translate it directly in english with "the whatever".