Or let my property manager so i stay anonymous?

Granted, there will be a large chunk of clients who value their privacy and anonymity when it comes to investing, but there should be a few who are willing to chat with you.

Have your cousin as the manager, while you anonymously be the sole member.

So if you want to keep it completely anonymous until closing that is fine. 

Ask questions they must answer via anonymous e-mail to get a reply from you. 

I also suggest a drive-by so they are not unpleasantly surprised by the neighborhood when they to the showing, and just keep on driving by anonymously as I wait outside.

You write that you got an anonymous notice that your yard needs to be taken care of.

If you are going for anonymity, then you need to create the anonymous-friendly LLC first, then make it the owner/member for every other LLC.If you are going the LLC asset protection route, then you need to run it like a business to maintain your corporate veil. 

We'll many times have the bank issue proof of funds in the account without any other identifier which can protect your anonymity. 

I also suggest a drive-by so they are not unpleasantly surprised by the neighborhood when they to the showing, and just keep on driving by anonymously as I wait outside.