Let me add to SwampeastMike's excellent response:
We will be arriving on our international flight by Turkish airline at 9.30 AM, how long would it take normally for immigration?
Allow at least an hour from touchdown to exit from the Customs area. It may take less, rarely more. As for a VISA, it depends on your passport. Here's the info:http://www.turkeytravelplanner.com/deta ... index.html
How far is the domestic section from international section?
The terminals are in different buildings connected by elevated and underground walkways. Allow 15 minutes for the walk from one terminal to the other, and anticipate a full security check when you enter the other terminal (as well as a full check when you enter the boarding gate). Here's more:http://www.turkeytravelplanner.com/go/I ... rport.html
Do the airlines generally run on time?
As much as anywhere, which is to say, sometimes they do, sometimes they don't.
Which is better airline for domestic travel?
As SwampeastMike notes, Turkish Airlines is the best. You'll even get a snack and drink on a 45-minute flight! But I have flown the newer airlines (Pegasus, Onur, etc.) with pleasure. Here's more:http://www.turkeytravelplanner.com/trans/Air/index.html
We are planning to fly Istambul - Izmir, Izmir to either Kayseri or Neveshir and then back to Istambul.
Does Turkey provide "visit Turkey" type of vouchers for foreigners to travel around?
Note that the non-stop flights between Izmir and Cappadocia (Kayseri & Nevşehir) do NOT operate every day, but only a few days each week. Otherwise you fly via Istanbul, which is more expensive and time-consuming. Here's more:http://www.turkeytravelplanner.com/go/C ... _capp.html
Have a great trip!