The car represents your body. Your sister, symbolizes one aspect of your thinking that is in control of your 'direction' in life at the time of the dream. Consider her personality. The way you would describe her is the same traits in you that was in control of your path. Would you say she was aggressive? Lazy? Uptight?
The crash indicates a negative event suffered in your body. This could be illness, injury, fatigue, etc. Basically, at the time of the dream, you were driving yourself in much the same way your sister thinks and you have or were going to suffer somehow for it.
The EMS represents a healing after the crash. Your dog represents a favorable habit that you have. One you didn't want to be separated from. (Even though the 'authorities' thought it best to take your habit away, you argued against.
Look over your life around the time of the dream and see where or how you were conducting yourself in a way that was similar to your sister's personality. See what kind of problems it caused for you and if you recognized the habit because of it.
Hope you can use this somehow despite it being late in coming.