Rent-a-car = Rent a car - simply a company title/advertising slogan with hyphens inserted into the sentence.
green-a-light [this makes no sense to me without more context]
Chef-o-matic - a typical advertising phrase/logo - it is combining the word "chef" [cook] and an abbreviation of "automatic".
Here we come a-carolling - archaic : Here we come, singing carols.
Student or Learner