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very interesting : from real-life problems to neural networks in a few lines !
How accurate are the iPhone's accelerometer and gyroscope?
If you mean how measurements of the iphone sensor are close to the real values, we did not test it and we do not have any equipment to perform such tests. For our purposes it's accurate enough.
Great work! This is really cool stuff.
This can detect an explicit lane change, but can't tell where you turned; if you make a right turn, it can't tell which lane you turned into. It would probably work on expressways with moderate traffic, where lane changes are usually made explicitly, one lane at a time.

You can do car navigation entirely from gyros, accelerometers, and wheel rotation counts, without GPS, if you have even moderately accurate map data. Etak had that working in 1985. I still have one of their original gyros.

Our purpose is to make a library that could be used by any driver without additional equipment, therefore we work only with the sensors available in a phone. Accelerometer and gyroscope installed in it are not precise enough to track position of car. Wheel rotation can't be measured of course by a phone.