Yes. Presuming you can identify the motorist, you should be able to recover for the actual cost of repairs for the damage - the motorist's automobile liability policy will be involved.
If the motorist doesn't have insurance, you can file a claim for the property damage done to your vehicles with your insurance company and the "collision coverage" provided by your policy; you can still pursue a recovery from him for damage to your yard but it is unlikely the motorist has anything to reimburse you for the repairs to your yard.
Yes, the standard auto policy contains liability coverage for property damage. In most car wreck cases, this coverage would be invoked to make repairs to damaged vehicles; however, it is also broad enough to cover damage to other personal property, such as "fixtures" on your property (like a brick wall), or even your house, or utility poles, guardrails, etc, etc, etc. Simply call the driver's insurance company and make a claim. You'll need to get estimates for repairs and provide the documentation.
You have not given any information to suggest that they would not be responsible.
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Yes. You should file a police report; the police department should investigate. You are entitled to compensation for the damage. If you know who caused the damage, his/her insurance will be on the hook. If you do not know, or if the person has no insurance, you should contact your homeowner's insurer.
Benjamin Andrews is licensed to practice in Virginia only. The preceding is for information only and should not be construed as legal advice. Always call a lawyer directly for advice.