That short ride will be fine. Just make sure the vehicle is not super hot or cold. Keeping the a/c cool but not cold should be good. Do not place the container near any vents. Remember most modern cars have floor vents so look close if you keep it on the floor.
Placing the snake in the new home is fine and yes we usually suggest not handling for 7-10 days.
The vet is up to you. If it is cbb from a good breeder then you probably do not have much to worry about. If its wild caught or from new wild caught parents a vet check might be a good choice.
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