Sometimes my character walks to a door or something and I spot these bags of gold. I try tapping them, but I'm always too slow, except for one time, but that was so close to the end of the movement or cutscene that I'm just not sure if I can pick them up.
Yes, you can definitely grab gold bags during cutscenes. In fact, some will only be visible in cutscenes. Also remember that you can find keys and potions lying around too.