You may want to provide more detail on what you are seeing or provide a screenshot to help visualize it. There is no viewable history of Airdrop files and no mechanism where a file that is airdropped can take control of your phone.
If you are seeing touch events on your screen navigating to different locations on your phone, that is a sign of a failing digitizer that is built into the screen. This occurs quite frequently when the screen is cracked, but of course it can fail at any time even when there does not appear to have a damaged screen.
There is no remote access capability on the iPhone. Your Apple Account can be accessed remotely if someone knows your password or another device is logged into your account. This does not give a person access to your phone, but the data that you see on your phone synced to iCloud can be changed or deleted. You can view if another device is signed into your account by going to Settings > [name on top], and scroll to the bottom to see the devices that have access to your iCloud data.