I ended up spending the night at the airport. After customs (it took me only 45 minutes to clear immigration and customs) I made my way upstairs to the departure area. Here I sat for 3 hours on an outdoor bench, playing with my iPad and reading a book. No one bothered me and it was great people watching. About 3 hours before my flight I made my way towards the exit. The domestic terminal is 5 minutes drive away from the international one, so I had to negotiate with a taxi to take me there. I paid $5. At the international terminal there was a good number of tourists heading to Khiva just like me.
Overall it was not a stressful experience - albeit very tiring!