Candor - I don't know the laws where you live, but in California, even a landlord has to give notice (I believe it's 24 hours) before entering your residence. What your neighbors are doing is illegal. Can you change your locks? I'd do everything I could to keep them from continuing that behavior, down to putting a lock on my patio and taking my hose inside so they can't use it. Post "No trespassing" signs on your patio door/gate. My neighbors have a camera in their front window, with a sign that says something like "Smile! You're on camera!" We had problems with people "casing" the area and breaking into cars. They installed the camera as a deterrent. I'd tell your neighbors that you'll install cameras and, if they are caught on camera entering your home without your permission, you'll go to the police.
I'd do some research online to see what the law says and then, if you can, have a lawyer send them a letter basically telling them to cease and desist. Just because you have a homeowners' association doesn't mean that they can behave this way. I'd also check into the law as far as your patio. Unless the law is different where you live, that's YOUR private property.
I only had a landlord enter my home once without permission. In my old apartment, there was a bad leak under my kitchen sink that I didn't know about because all the water was going underneath the cabinets and then into the apartment below mine. They had to enter and fix the problem while I was at work because the water was causing damage to the unit downstairs. Your window being open is not grounds for your neighbors to enter your home, IMO. If there is damage, it's none of their business, as long as you fix it before you move.
Just reading your post about all this has gotten me riled up! I'd be furious!