A lot of times when I'm just moving my phone around, it will switch into the mode where you can add/delete home screens. Is there a way to avoid this? No idea why it would even do that. I'm not even touching the screen. :-\
The Note 5 has such slim bezels along the sides, unless you're using a case that has a sufficient border along the edges, merely gripping the phone on the sides can often initiate an action. Maybe that's what's going on.
Does the TW launcher have the option to lock the homescreen? I don't use it so I don't remember. That will help prevent accidentally long pressing the homescreen.