All MOO, IMO, etc....
I'll eat my hat if this is not premeditated. First, he lures her with the fax (while her mom/his wife is conveniently out of town), and promptly shoots her in the head. Right after that, he sends the fax as "proof" that she was there and the task was completed. Then he concocts some story for LE when they arrive to question him about the fax about how he was not there when she arrived, that he was out doing another estimate, but oops!, the homeowners weren't there, so he just does a brief rundown and takes a few notes while he is there (no surveillance cameras at the house either, just like the dollar store parking lot). He has a "pat" alibi, he thinks, and performs it perfectly for LE. He isn't acting "quite right," but LE knows people act weird in stressful situations.
Over the weekend, LE interviews more friends, family and acquaintances. It's still too soon to move the body as LE activity is at a peak. On Monday is when LE pulls the surveillance (see page 1 or 2 of this thread for the MSM reference). And, well, whadda you know? There he is dropping off his truck near a mechanic's shop right by the dollar store! Funny, he never mentioned that trip or occurrence that day. So, LE goes back and re-interviews him giving him the chance to own up to it voluntarily, except he doesn't. So, they pull that little gem out of their hat and he knows that they are on to him. He then gives the bogus story about his truck needing repairs (quick CYA). LE calls the mechanic and nope, he didn't talk to the shop about his truck. His alibi starts to crumble. LE goes out and talks to his neighbors who may or may not have reported seeing him during times he was allegedly gone from the house. It's then that they have enough evidence to get the search warrant.
I don't think he has talked much to LE. It sounds to me that LE is inferring it was sexual in nature due to the evidence (and some not reported right now), not anything he said directly.