Q:

Do Door of Destinies and Myr Turbine interact? 

If I have a Door of Destiniesin play and choose creature type Myr, do the abilities of Myr Turbine interact with the Door? 
Do either of Myr Turbines effects cause Door of Destinies to trigger?  Or, Myr Battlesphere.  Do the 4 myr tokens it generates count? 
Seems like the second would but im not sure about tokens...

A:

If you choose Myr as the creature type for Door of Destinies, all Myr, whether cast from your hand or searched for or tokens, get +1/+1 for each charge counter on the Door. 
However, the Door only gets counters when you cast a spell of the type Myr.  What you are doing with the Myr Turbine is activating an ability.  The first ability puts a token in play, the second a creature card from your library, but in both cases it's activating an ability, not casting a spell. 
Casting the Myr Battlesphere (and I mean casting it from your hand, not putting it in play using the Turbine) would activate the Door of Destinies, but only for the Battlephere itself, not for the subsequent tokens because they don't get cast but result from a triggered ability.