Khiva to Urgench for 8:40 am flight

Your best bet is going to be shared taxis. I've taken a taxi from Khiva to Urgench, and the problem isn't finding a taxi, it's finding one that is almost full.

The issue is that it's a big culture of shared taxis there. You negotiate your seat to Urgench, and then wait for the other seats to be negotiated. When it's full, the car drives to the destination. I did this in Uzbekistan and Tajikistan. Kazakhstan and Kyrgyzstan seemed to be more keen on regular one-person-per-taxi.

I'm sure, however, that you could negotiate a rate for the whole car if you wanted to. It's not far and they'd get some money out of you. It's about 35km from Khiva to Urgench. From memory, taxis wait outside the north gate of the old city of Khiva (on your left as you walk out), although I'm sure you could ask someone to order one for you - the information center by the museum offered to do it when I was there three years ago.

Wikitravel is claiming that the going rate for the whole car for that distance is around 8000 sum, but that'll depend on the current rate, when that was written, and how much the drivers think they can get out of you - make sure to negotiate, or get someone to do it for you :)

Regardless, 35km is not too far - but the roads aren't great. Allow an hour to get there. Even allowing that however, you'll then need to either get another cab, or convince your driver to take you to the airport. It's definitely possible, it's just up to you to decide on risking detours, punctures or delays.

Good luck, Khiva was my favourite town in Uzbekistan!

We stayed here at Hotel Arkanchi when in Khiva a few years ago. Not fancy, but great location and friendly crew.
And according to the website, $35 a night for a single person, or $55 for a double room with 2 people.
Service we had was good, might have helped with had a guide who spoke the language and didn't have to rely solely on hoping to catch an English speaking native.

From Urgench, hour or so. Of course waiting for a taxi (as already mentioned) may add considerably to that. But there's quite a few large hotels in the area so depending on time of year you might well get lucky.