I will answer your first question. To desert is to run away from an obligation, which is why to desert one's car is not the best answer given the more precise semantics of abandon (i.e., a person or place or thing, such as a car).
desert (verb) to leave (a person, place, etc.) without intending to return, esp. in violation of a duty, promise, or the like: He deserted his wife.
Source Desert Definition | Definition of Desert at Dictionary.com
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