They did remove it, but I have changed property management companies, and now, 2 years later, I see that the trampoline is installed again, and the tenants have installed a second shed without my permission.

You do not need Airbnb permission to do so. 

Gordon Ramsey has had a restaurant for sale and it was taken over by professional squatters (didn't know that was a job title :-)They planned to give away free food and open an art gallery there and are threatening to prosecute anyone entering without their permission.

But I think some people see that and think it's permission to take advantage.  

NEVER from previous experience yet some will do it regardless of permission. 

Or you can do a "permission to show" to represent you investors, and still collect a commission.

Get someone to stay in there before more damage happens, security lights, a doberman.Please grant permission to use that medicine cabinet photo. 

If you are interested in an area, look into renting a LTR from a responsible landlord and, with their permission and clearly defined (in writing) roles and responisbilities, then sub-lease their unfurnished rental as an MTR once you furnish it and get it ready.

I spoke to local owners who have been audited for increasing rents without there permission and they had to pay back the difference!

You can also have your cleaners download the app and have specific user-permissions to allow for task-based scheduling and cleaning checklists.