hi,
http://www.tuvaavia.ru informs us there are no flights Irkutsk-Kyzyl for 2010.
Actual timetable is as following:
SL 2037 Novosibirsk - Kyzyl 08.25 11.35 1....6. 25.10.2009-27.03.2010
SL 2038 Kyzyl - Novosibirsk 18.30 20.00 1....6. 25.10.2009-27.03.2010
SL 2039 Kyzyl - Krasnoyarsk 13.50 15.20 1....6. 25.10.2009-27.03.2010
SL 2040 Krasnoyarsk - Kyzyl 16.10 17.40 1....6. 25.10.2009-27.03.2010
So from Krasnoyarsk or Novosibirsk only. Monday and Friday only. The airplane is An-24 small and noisy.
They have contact phones: +7 39422 22064, +7 39422 22214 but I wonder if they speak English.
For YI the flight Krasnoyarsk-Kyzyl costs about $150. Try the itinerary Irkutsk- train to Krasnoyarsk- flight to Kyzyl. And return way Kyzyl - flight to Novosibirsk - flight to Europe
Also you can go Irkutsk - Abakan by train and then take taxi to Kyzyl (that way takes more time)
All russian trains are here
http://eng.rzd.ru