Do Door of Destinies and Myr Turbine interact?
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.