Strange behavior of Adafruit motor shield (not original one) and DC motor

You've probably meant 7.5V?

I didn't think before about too much current draw since that power scheme was taken from the "Make an Arduino-controlled robot" book.

Regarding capacitor - I shall add it across batterys terminals, right?

Power supply - I have 9V DC adapter, I could try it out - plug it in arduino and make shield use arduino's power source.