Will steroid inhalers affect my growth?

Probably not. The first generation of inhaled steroids on average shortened growth 1 inch (but untreated asthma shortened growth quite a bit more than that). The new inhalers don't shorten growth even that much. Out of control asthma, however, is deadly and dead people don't grow.
Not likely. While short-term studies suggest that inhaled steroids at higher doses may affect growth rate, long-term studies indicate that persons using these important asthma controllers grow for a longer period of time before reaching final height. So they catch up in ~6 months. Who to believe? Worst case scenario is 1/2" shorter than without inhaled corticosteroids. Hundreds of my patients grow just fine.