Hi!
Some answers here
1) Itinerary: 3 days for Bukhara, 2 for Samarkand, 2 for Khiva. Is that OK?
Yes, it's quite OK for these cities. Just need a guide or perfectly planned Google map;
2) Should I spend time in Tashkent?
It's up to you. As all of Asian capitals it's big, more or less modern city, a mix of old city, soviet architechture and modern buildings. I think couple of days may show you nice places.
3) Any need for vaccinations?
No :) Just take couple of sanitizers and wash your hands :)
4) Transport between the cities:
Do I understand correctly there are flights only from Tashkent to other places, but not directly between the others?
Yes, but better to take 1) Train from Tashkent to Samarkand (very nice one), 2) taxi from Samarkand to Bukhara and 3) taxi or train (old one) from Bukhara to Khiva and then 4) fly from Urgench (near Khiva) to Tashkent.
How much money and time does it take for a private taxi from
Better to start with Samarkand, IF you arrive to Tashkent. As Samarkand closer and there is good train Tashkent-Samarkand
Tash to Bukh (to Bukhara it's about $50-70 per car, depending on car and can be shared among passengers.)
Bukh to Samar (about $30-40 depending on car)
Samar to Urg (about $50-70 from Samarkand, and about $40-50 from Bukhara)?
Have a look at map, decide final route and I'll try to get more exact time and prices
What about the train option: reliable, comfort, prices?
Here are links to official Railways website:
Schedule - https://translate.google.com/translate?sl=ru&tl=en&js=y&prev=_t&hl=ru&ie=UTF-8&u=http%3A%2F%2Fuzrailpass.uz%2Fservice%2FrailRun&edit-text=
Prices (in Uzbek Soum, roughly divide by 4000 to get USD) - https://translate.google.com/translate?sl=ru&tl=en&js=y&prev=_t&hl=ru&ie=UTF-8&u=http%3A%2F%2Fuzrailpass.uz%2Fpages%2Fview%2Fprices%2F&edit-text=
5) Pls recommend guides for these three destinations.
Not sure abuot guides, can advice more specific things and take you in tour in Tashkent :)