How many universes do you think there are, and why?
I say one, because the universe technically contains ALL matter and energy, and there cannot be matter outside the universe. So yes, NO EDGE.
One. Thus far, all the evidence I have seen for a multi verse has been speculative.
It's hard to say how many. The WMAP satellite's heat pattern map of the observable universe has shown interesting patterns of hot and cold. Further, the super galaxy cluster near the constellation near Centarus A is moving incredibly fast towards it, suggesting that something massive may be near by. One possibility is that this is another universe bubbling up particularly close to our own.
Here's a video on the subject: http://www.youtube.com/watch?v=dG1JpC5jels
As for whether the word "universe" and "cosmos" are synonyms, Dr. Brian Greene's TED talk on string theory may present more information for you to make the jump to the multiverse: http://www.youtube.com/watch?v=yi4A12nHyo4
They're quite long, but if you're interested in the subject, they are also quite good.
You should read the Many Worlds theory. It's a little far fetched but it explains the possibility of multiple universes.