Detecting the presence of a specific smartphone
Can I easily detect that a specific smartphone is in proximity ?


It can be done, yes. How "easily" depends on how much money and expertise you have to throw at it.


  Can I use the smartphone's NFC (RFID) to do so ?


It is possible to use that, yes - if you add the right hardware and software to the Arduino.


  How long can an Arduino run on battery using a 9V battery from the market ?


A couple of hours if you're lucky. They are useless batteries.


  What kind of output could I possibly get from an Arduino ? I would like to make statistics, so it must be stored somewhere, and I know Arduino has not a lot of memory.


Whatever you want. Again, it's SMOP / SMOH.


  Is it possible to write on an USB key ?


Sure, if you get the right hardware for that.


  Which Arduino is best suited for my needs ? Should I consider buying an Arduino with Ethernet in order so send email ?


The Arduino that provides the facilities your project requires - and that depends on how you are going to decide on how to do it. Remember: the Arduino is just the "brain" of the operation. You need to decide what "organs" the brain needs to control and in what way. 

There's a million and one ways of achieving what you want. You could even do it 100% in the phone in software (there's probably an app for that...) without need of an Arduino...