Technically, because Duo is using the feminine form année you have to use cette. The larger question is why année instead of an? Well, for demonstrative adjectives (this/that/etc) you would normally use the masculine an, EXCEPT for this year which must be cette année
BTW, on the audio exercise: cet and cette are pronounced exactly the same: /sɛt/. You know that is has to be cette for the same reason: année is feminine.
See Lawless explanation about masculine/feminine time words: http://french.about.com/od/vocabulary/a/an-annee-jour-journee-matin-matinee-soir-soiree.htm
The answer for that is in the same link, In summary:
cette année == this year
cet an == that year
Except demonstrative adjectives, which are used with division words:
cet an - cet an que j'ai vécu en France
that year - that year that I lived in France
(But when talking about the current *year, say cette année - this year.)
Yes, but to me this is a good thing. It means a lot of new material is being presented. I know of I stick with it and i don't get discouraged, soon it will be easy.
Actually, I'm on the last levels of this unit on the passé composé. A few days ago these sentences totally baffled me. Now, after much repetition, they are truly easy. I now understand the verb agreements, which a short time ago seemed impossible.
I'm very happy with the progress I'm making, and I hope you'll be doing well also. The naturalistic teaching approach does mean there will be periods of confusion, but persistence will pay off!
It is avoir, I have not....That's all. When the course progresses to past tenses, a quick dip into French.about.com/passé composé (which ties to past imperfect) will give you a quick and simple lesson and access to other tenses,- and a brilliant conjugator that will also work back to the verb (if you did not recognise vécu, for instance).
New tenses are a bit of a leap, so they require some work. But once you have 'got it', things get simpler. The link above should really help. I am a fellow learner, I hope this info has helped.