Yeah, DuoLingo's approach to teaching grammar appears to be 'Just go with it until you figure it out'. 'Avez-vous' is used to ask a question; in French, to ask a question, you reverse the order of the subject and the verb, and then add a hyphen in between. So 'vous avez' become 'avez-vous'. There are other ways to form a question, but this is the formal way (like the difference between singular 'vous' and 'tu').
To help the lost get a sense of the idea:
'Mangez-vous une pomme ?' Are you/do you eat an apple?
'Aiment-elles le jus ?' Do they (feminine) like juice?
'Est-il rouge ?' Is it (masculine) red?
if you change the order of the subject and verb it automatically becomes a question. "Vous avez un chien" can be a sentence or a question but if you want to make it clear that its a question you can change it to (verb)-(subject) ie "avez-vous un chien?" for example i could say "tu es un homme" (you are a man) as a statement or question or "es-tu un homme?" which makes it clear that I am asking.
Gotta point out here that "gotta" is used as an informal shorthand for "got to [do something]" so your example would be better as "got a dog?"
Unless, of course, you were asking someone if they have to dog. Where dog is a verb. I don't know, some people might have to dog every once in a while for all I know.
There are a number of ways to form questions in French. You will be exposed to them all in due time. If you just can't wait, here's a sneak peak: http://french.about.com/od/grammar/a/questions.htm
The pronunciation is chien. When you listen to the turtle version it is more prominent. She says something like "she-on"
The pronunciation of chienne is something like "she-in" It is a little tricky. Kidding, it is so tricky, if you are having problem with pronunciation you should listen to the pronunciation of chien and chienne in a translator. Or you could go on http://www.forvo.com/languages/fr/
You are no doubt referring to the French "inversion" method as one way to ask a question. Instead of saying "Vous avez...." (You have....), you invert/flip them and put a hyphen in between. It becomes "Avez-vous...." (Have you....? or better, "Do you have....?") http://french.about.com/od/grammar/a/questions.htm
When an s makes a liason (what it is called when a letter from the word becomes pronounced and attached to the word after it that starts with a vowel. In this case, that letter would be the s in vous), it creates a z sound. Son has an softer s sound. "Un" and "on" also have different nasal sounds; "un" is more in the nose and the lips are spread apart, "on" makes the lips form more of an o shape. Differentiating the sounds usually takes time though. Most people have to get used to French's overall sound before they can start picking apart the more foreign sounds. That will come with time and practice.
Interestingly, I got this one correct even though I guessed at the spelling of the first part of the phrase. I spelled "Avivous" instead of "Avez-vous." it gave me a correct response and just said to me that I had a typo. This goes to show you that if you make a guess that sounds correct you may actually get a positive response from duolingo
I have started reading the tips to start every lesson and I end up reading the same tips many times because it takes me a while to get to my goal of level four before going to the next level. This has really helped me so much to understand and absorb the tips given and to be able to apply them. I often see questions in this form that assure me that the student has not read the tips section. It really is a huge help. I highly recommend reading the tips.
I repeated this exactly as the system said it - and it was rejected - WHY? This is a continuing EXTREMELY frequent problem, and very disheartening. And does this ridiculous childish character Duolingo (or anyone who works there) EVER read these discussion points - to learn? It's time Duolingo learns what users think about it.