I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear that you're having so much trouble with your neighbor.
Your neighbor had no right to remove your sprinkler system if the sprinkers were, in fact, on your side of the property line. You have the ability to sue for property damage (for the cost of replacing them), plus trespass, which would impose punitive damages on your neighbor for entering your property without consent. The judge can also order them to pay for removing the sprinkler heads and restoring your property to the way it was before (filling in the holes - get an estimate for a gardener). Any litigation attorney can help with all that.
If the HOA didn't take any negative action against you for teaching piano lessons out of your home, then there's no cause of action against the neighbor for saying you did it. This is both because teaching the piano is something that you do and because telling people that you do it isn't likely to harm your reputation. But you may be able to file defamation against her for telling people you're unfit to live in your house, if the allegation goes beyond APS (because those statements are protected), or if you can prove she knew the allegations weren't true when she made them. Trying to get you committed because she doesn't like where the property line is could also be considered intentional infliction of emotional distress - it would depend on whether she made one call or there was an ongoing pattern of activity. A litigation attorney who does intentional torts could also help you determine if there's an additional cause of action here. You might also be able to sue for nuisance, because your neighbor is interfering with your use and enjoyment of your property.
A member of an HOA board has an obligation to do what's in the best interests of the entire community. If the president is bending the rules and assisting a property owner in harassing you, look at the bylaws and rules and regulations to see if there is a procedure for removing that person for violating their fiduciary duty to the community. Another option is to document everything that happens, and how this person is involved, and bring it all out into the open at the next election.
This isn't exactly a legal solution, but once you get things worked out in a court and a judge decides where the property line is, it may help to put up a fence, both so you can ignore each other for the most part and so your neighbor can't come onto your property and mess with your landscaping/sprinklers again.
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