In my opinion, the short answer to your secondary question is "both." Shakespeare intended to use quibbles and puns and those forms of speech were common in teen vernacular of the time. In order to illustrate my point, let me report a very short exchange that contains more puns and twists than any I have ever seen:
Romeo. Good morrow to you both. What counterfeit did I give you?
Mercutio. The slip, sir, the slip. Can you not conceive?
Romeo. Pardon, good Mercutio. My business was great, and in such a case as mine a man may strain courtesy.
Mercutio. That's as much to say, such a case as yours constrains a man to bow in the hams.
Romeo. Meaning, to curtsy.
Mercutio. Thous hast most kindly hit it.
Romeo. A most courteous exposition.
Mercutio. Nay, I am the very pink of courtesy.
Romeo. Pink for flower.
Romeo. Why then is my pump well-flowered.
Mercutio. Well said! Follow me this jest now till thou hast worn out thy pump, that, when the single sole of it is worn, the jest may remain, after the wearing, solely singular. (2.4.47-64)
There is simply so much contained in this simple conversation! The beauty of it is that I have seen it played quite serious indeed (and sometimes even with the most raucous bits left out). Of course, I am more a fan of the bawdy interpretation. The puns are incredible, depending on how you take the word "slip" and "business" and "bow in the hams" and "pink" and "flower" and "pump" and "single sole." Many a lady, I suspect, has swooned in thinking about Romeo's "pump." Ha! This is also a very common exchange that critics point to as evidence of a physical attraction between the two friends. In my opinion, the beauty of this scene is that these off-color jokes are just the kind that teens spout in the halls today, although not in Shakespearean language. Some things never change. That is the one of the many reasons that the genius of Shakespeare has stood the test of time.