I have just returned from Uzbekistan, and would only recommend your plan if you are fluent in Russian, Uzbeck and Tajik languages.
I used the train, shared taxi and one domestic flight to get around.
It enabled me to mix with locals, who were keen to practice their English skills, but ensured that I could not go too far off the beaten track and get myself into any kind of trouble.
A shared taki took me from Buchara to Khiva and a flight from Urgench back to Tashkent.
I would't know about any car rental company at Tashkent airport like we normally have at any airport in Europe. The second thing is that in the city of Tashkent for example, you would get lost in the first five minutes. No road signs at all! Traffic lights are positioned in a way that you have to understand sometimes.
And if you need a car in Tashkent all day, you pay not more than 20USD. And you will not be in a constant stress, because of surviving traffic!
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