Jack had been without a car since Monday.

In sentence 1, the word since is a/an

What are you asking, exactly? Are you asking if the word is to be a or an? Because it would be a car, not an car, because it makes the sentence flow better and sounds correct. Usually, in english, if you can read a sentence and it sounds correct, such as "The dog barks", it will be correct, whereas a sentence that sounds incorrect will most likely be incorrect. Such as " Barks, the dog".