Why didn't the hospital go into lock-down? (When shots are fired, the place goes into lock-down, basic policy, NOBODY's just walking out invisible or not)
Emily knows that Adrian is abusive and had influence over other people. He could have easily set up someone to send her a fake email. Typing is universal and impersonal. You don't just automatically believe that if you're willing to stick your neck out for the person.
Wall paint doesn't wash off with water (Wall paint is oil or latex based. This means you need to use cooking oil or rubbing alcohol to get it off even when wet. He'd be easily detectable)
We see the sink filled with wet paint. Even if that would work, that suit has many nooks and crannies, try washing off paint, it takes LOTS of scrubbing to get it off. It's NOT a fast process.
He didn't notice his other suit is missing (He's supposed to be following her, he had into their old house, knew she was there, but doesn't bother to check what she was up to?)
That kind of suit is not something to just be left lying around. He followed her into the closet before she assaulted him. A good search of the closet would have revealed she took the suit and was hiding it for nefarious purposes later.
Cecilia was just getting up, she didn't reach out or strike. Anyone could see that! Jumping to that conclusion is just foolish. More than likely she'd wonder what just happened.
He's INVISIBLE not UNDETECTABLE (No creaky floors, no footprints in the sand/carpet, walks through a metal detector and doesn't set it off, only one dog in the whole movie)