1. Yup you have to give the bike a once over on delivery as it appears they are being rushed out at the moment.
2. Check the battery holder, the parts that hold the battery are adjustable (have a feeling it may only be the bottom one but might be both ends), to make sure there is a tight fit - if the battery is slightly loose, or one of the connector loose, this would cause bike to turn off.
You can check the cable connections by popping off the motor cover on the LHS.
The bottom battery holder, which has the electrical connection, is designed to have some side to side movement, so don't think this isn't normal - its to help get the battery in.
3. Yes this is fairly common - the head unit is calculating your range/battery remaining off how you are riding at the time, so if you are caning it, it will predict less range etc, whereas the battery indicator seems to be more linear. As you have noted the display can show one bar, and the battery two, but if you switch off then turn on again the display will go back up to two. This is because you have effectively reset its memory of how you are riding, and commenced a new ride.