Yes frequent and high dose steroids cortisol, prednisone can effect height, usually also with associated weight gain as well as slower height gain. You should set up a followup consultation with his physicians and especially check that they have actually PLOTTED height and weight on the growth charts of his medical records. Could be unrelated to thesthma or steroid treatments and might be a family pattern as well as along list of other possibilities for delayed puberty and slow growth. Full detailed history and complete physical exam including size and shape of penis, how much and what Tanner stage are his genitalia pubic hair, testicular examination and size all will help in this assessment. If still unclear explanation, then some lab screening tests like urinalysis, blood count, sedimentation rate, Creactive protein level, full chemistry panel, thyroid tests T4 and TSH as well as IGF1 screen and hand bone age xray determination may also be needed based on his actual plotted growth chart and exam. If still persisting, then consultation with a pediatric endocrinologist would be recommended.

That is possible, that some children are more sensitive to corticosteroids than other. Some of my patients benefited from non hormonal medications like Singulair and non steroidal inhalers. Ask for an allergist consultation to identify and control the allergy.

Yes, prolonged steroid use can affect growth, including height (slows down) and weight (might increase). The specifics vary from person to person depending on how the steroid is administered, its potency, and its duration of use. It's important to discuss these concerns with your doctor who prescribed these first before taking any action, since the medication may be critical to their well being in spite of this potential risk.

Corticosteroids are immune modulating medication. Their use in controlling verious inflamations makes them very important medications. We use them for asthma, and other inflammatory disease. However they also have side effects. They could make us gain excess weight, suppress growth in children, cause osteoporosis and more other side effects. We use them under many circumstances because their benefit outweighs their adverse effects.
Some inhalers for asthma are corticosteroids. They are important part of the treatment, though they could stunt growth and cause other side effects.
If your child's growth is suboptimal talk to your doctor if the inhalers could be changed. But if the steroid inhalers are more important to control the asthma, you need to discuss with you doctor to see an endocrinologist and explore the possibility of hormone therapy for his growth while he continues with his steroid inhalers.
Good luck

It may decrease his growth velocity and decrease his final height

Yes it is possible that steroids are impacting his height. Please do not stop the steroids without discussing with the doctor who prescribed the steroids.