My guess is that you are checked out now, but if not, you should respond to one of the daily emails by explaining that you are properly caring for the apartment. Then state that, "We are caring for the property. Please, under no circumstances should you enter the property while we are leasing it without our presence or permission. We will be glad to water an plants. Please do not invade our privacy. Thank you."

I'd be willing to bet that these people just put their property in the rental program and are paranoid. They either need to chill or quite renting. And I agree with the other posters that after you get your deposit back you should blister these people with a review on the website.