Lambert sensed Cole in Evas room where Eva only slightly noticed when Cole made a sound. There wasn't detective work then, and he had no reason to believe anyone else would have been in there with Eva originally unless he's like that with every room he walks into. Which would be a funny image to imagine Lambert stepping into a room and then staring at the corners for five minutes each...
Cole only started murdering people a year after he met Rhys after the Kirkwall event like I said. It probably was blood magic/spirit/demon magic he was doing when he was feeling the 'pull' to mercy kill those people, or it could have been the spirit/demons left over remorse of being unable to help Cole having seeped in. Cole at the end of the book pulled a distressed face when he mentioned being unable to help the original Cole in the dungeon.
And what if Cole hadn't actually been in the tower for that long? Cole said he didn't remember when he got there, the Ghost of the Spire rumors where never established to have been there for years considering that it didn't seem to be that common to the others. The way it's worded in the book it sounds like it could have just started up when Rhys first saw Cole in the lecture hall, Rhys had been there his whole life and had never heard about the rumors of the ghost until he saw Cole? Cole could have been there for not that long but had already quickly learnt the layout of the areas below (after all, he's not human and wouldn't have a human level of memory, and he was 'quick to learn') For all we know, Coles actually only been at the tower for a year, since Cole had no idea how long he's been there and Rhys has only known him for that year. So his weakening feeling could have been a years worth being in the physical realms taking it's toll on him.
All theory crafting here, but it's been bothering me for a bit why Rhys wouldn't know the Ghost of the Spire tales having grown up in the tower for 40 some years.
Cole may have been unconsciously making people forget him, but he had to focus a ridiculous amount to get it to the point where he could kill a person without them realizing it, so the idea of Cole controlling Rhys to that levels makes zero sense. Not when Cole was using his magic to mask Rhys, Rhys felt it and knew it was being done on him. Not when Cole has such little control of himself. Not when the idea of Rhys being controlled to stab someone really difficult for Rhys to do when the circle was on lockdown after Kirkwall, stabbing a person up close would leave blood on his robes and a dagger that is never shown in Rhys hands until the very end of the book. If Cole was there to clean up the mess, then it wouldn't be unconscious levels.
Why would Gaider write the book through Coles point of view if it's completely unreliable? Gider never struck me as a writer who would have the 'he was dead the whole time and everything around him was a lie' twist, certainly when there's no breadcrumbs for the reader to come to that conclusion.