Ok guys, I have an admittedly crazy theory. But just follow along for a minute....
Part 1: Random attack
For starters I was considering scenarios that would have to be motive for a random attack, and concluded for a random attack, it has to fall in some way into one of these categories.
1. Robbery
2. Rape
3. Psychotic break by disease or drugs
Looking at the robbery angle, from the crime scene I just don't see it. Why would certain things be left and why would certain things be taken? Why would her pants be off? In an act of robbery why wouldn't they just knock her out and take her stuff, and leave the body? If robbery was the only reason for this crime they could have gotten away with it much cleaner and wouldn't have gone through the effort to conceal the body. Robbers typically do not like to also be murderers, so if sarai put up such a fight over her stuff, I see a robber just taking off.
As far as rape, though her pants were removed, we haven't heard any other evidence of a possible rape. I'm not 100% positive but I am pretty sure the other girls raped at that spot were in the evening, but if someone else remembers better please correct me. I just feel if someone wanted to rape a 5'2 110lb woman, and they had raped before , they would have succeeded.
As for the psychotic break, it would definetly explain the disorganization but I don't think it would explain going through such an effort to cover it up. Especially making sure her earrings, and anything else that was dropped, including her license were with her. And I don't believe for a single second that someone killed her, and then someone else came across the body and took her stuff, and then someone else hid her. To me at least the idea that people would come across a dead foreign woman's body, and rob or move it, is ... Well I dunno I just don't see that at all.
Ok so now lets look at Z. What do we actually know about Z? We know he chose to live on the streets, we can assume this was do to a mental disorder but it does not mean this is necessarily so. You see though schizophrenia accounts is much higher, percentage wise in homeless populations, then in non homeless population it does not mean all homeless people have schizophrenia.
Based on some of the things we have learned about Z's home life, I'd be more inclined to believe that any mental issues Z had in life, were more mental disorders learned from compensating or dealing with the life he had, opposed to organic and biologically based mental disorders. We learned his father passed away when he was an infant, that his stepfather was harsh to him, that he didn't start school till he was 10, and only lasted 3 days before taking to the streets.
For a child to not start school till 10, and then be out in a classroom, either with children significantly younger, or children his own age but more educated, either one would have been emotionally stressful, extremely stressful, a huge blow to his self esteem. I'm sure he was laughed at for not knowing things, for possibly being illiterate. It doesn't surprise me at all that 3 days of school like that, and in all likelyhood an unhealthy relationship with his stepfather, would cause a 10 year old boy to flee to the streets. He probably felt like he had more street smarts, then school /society smarts....he wasnt being made fun of on the streets, he had found a place where he fit in. This to me seems more like an accurate picture of why Z took to the street over a biologically based psychosis of any kind. He was tough, he was hardened, he had a shell, he learned not to trust anyone but himself, all aspects that lead to compensations that can be classified as various types of personality disorders. But I don't see schizophrenia. Another small reason I don't think this, is because he was in the Turkish military for a while (which is I believe a 2 years requirement) for ALL males in Turkey, which I believe has to be fulfilled by the age of 33 (?). (Almighty or Yashim might be able to confirm the required time to serve and by what age better then I can). Anyway schizophrenia has a typical age of onset around 18-23, or early adult years. I just find it difficult to believe that a completely schizophrenic individual would be able to fulfill this commitment. Also some of the things we have heard about an hr long chat in the coffee shop in karabuck, and normal behavior on the bus, just don't paint the picture for me of someone with a severe psychosis.
Finally I was thinking what actual evidence is there really of Z at the crime scene? From what we have heard that his DNA was found under her finger nails. To me all this really tells us (assuming of course that it wasnt planted) is that Z had contact with Sarai, contact that she didn't want. But to me it doesn't really tell us he was the murderer. What if he was just the one who was restraining her, the only one close enough to her hands to reach to scratch. It really doesn't tell us he killed her. It just tells us he had contact with her.
Ok.....crazy theory coming up in part two...