How much can falling down due to a stall influence the angle of attack?
In AF447 the falling motion most definitely affected the angle of attack. Due to the fact that the pf did keep the stick pulled back for all but the last few seconds of the fall the aircraft was so steep that, although the aircraft was falling at a slight pitch up, angles of attack actually reached as high as 60°. 

The descent was so steep that the air was passing over the pitot tubes at such an angle that it was reading below 40 knots, although the plane was moving far faster then that. The computer required that the airspeed be of a certain value to consider the aoa plausible so the stall warning stopped sounding even though the aoa was between 30° and 60° during the fall. When it hit the water the forward ground speed and the vertical speed were both 107 kts. 

As far as why the pilot did not realize he was stalled, that's the question of the century. Much effort has been put into figuring that out, and there are numerous things that are at play. But in the end I don't think anyone has come up with a satisfactory answer for that. 

If you want to dig into the causes of this crash and get a good understanding of Airbus systems in general I highly recommend this book and it's companion website. The author is uniquely qualified and gives the most thorough explanation yet is not as technically challenging as reading the BEA report. 



All of the numbers mentioned in this answer come from the linked book.