4.) The seller will most likely give the animal in some container but I do like taking my own. I just take a locking lid tub and a towel. On the way home I just make sure the cab of the vehicle is not as cold as I would usually keep it.
Being you are asking about a BD I am not sure what temps they need. As long as you can stay close to acceptable numbers I would guess they would be fine just like a snake.
If you have heated seats you may be able to use those to help. I have before during cold months. I do always take a thermometer so I know the temps though. I keep the seat on low and watch it.
Having your kids with you they could also simply hold it on their lap and that should give the needed heat.
Just spitballing.