Sorry to say this again but I'm still really stuck on this shoe thing. I believe in the surveillance she was mostly wearing boots. I doubt she would wear shoes that could fall of easily, because it was winter and it was cold, and she would have been walking a lot.
Assuming she wasnt wearing shoes that could have fallen off easily...and it was reported her pants were next to hear body. Then in order to get her pants off her shoes would have had to be taken off. But I haven't heard any mention of her shoes being found at the crime scene, or if they were where they were located.
But let's just explore a theory for a second....I don't mean to offend anyone, nor assume too much about Sarai, just trying to look at this case from many different angles.
What if Taylan and Sarai had gotten very close even before she came to Turkey. What if there relationship was very flirty. What if because of the supposed issues in her marriage, Taylan had become the person she confided her deepest feelings in, even feelings she didn't share with her best friend back home. What if she felt her world back home was too interconnected and she couldn't be honest with everyone and she was just trying to be strong for her boys. So lets just assume for a moment there is some truth to this. As they get closer he starts saying he wants h to come to turkey and she wants to meet him too. The idea starts small, but grows bigger. He suggests many reasons she could come, and tha she could even explore her love for photography in Istanbul.
So suppose she comes to Istanbul and she meets with Taylan and its flirtatious and fun and amazing (she has been with the same person since high school, so someone new showing her all this attention is heaven). He sweet talks her, he even tells her he loves her, Turkish men are notorious sweet talkers, and they like to use the love word early. Maybe they have even made out a little.
But during this time, she starts to miss her husband because she realizes their bond is so much more important to her, and she becomes homesick. She doesn't tell this to Taylan.
What if they were there together at that spot, taking pictures, being very private. What if the even started fooling around. And in this she stops him and says there is something I have to tel you...perhaps he has been nothing but sweet to her up to this point. She tells him she is leaving that day, he starts convincing her she should stay, they can go anywhere together, he will take care of her. She keeps shaking her head no.
His sweetest starts to turn to anger, she sense it, she tries to walk away, he grabs her, she pulls back the whole thing escalates very violently, winding up with her pinned down and him grabbing the nearest thing to him, a rock. He goes into a rage.
When he stops he realizes what he has done. He's smart so he knows he has to conceal the body. Maybe he takes the jacket because his own blood got on it. Perhaps her shoes had already been off because they had been intimate. Perhaps her feet just hurt and they were sitting on the blanket chatting and she took them off.
After he conceals the body he takes the iPad and cell phone to erase the fact they were planning to meet, or any conversation that insinuated they were more then friends. He has no interest in her jewelry. But he takes her back pack to carry the electronics in. Maybe he took her shoes to hide finger prints, he might have been able to put them in the backpack.
I'm not saying this is what happens just one of the many many theories we have all had.
If this is not the story or something similar with another online friend...my best guess would be that the whole crime scene was staged to look like it took place there.