Can being on steroids for 5 days cause permanent behavioral damage in a 3 year old?

Not likely: Steroids are most commonly used got their anti inflammatory effect for asthma, allergies etc. They should not affect the behavior in a child especially such a short course for 5 days.

Unlikely: Unfortunately, the dose of steroid needed to help with various conditions in kids can set them into orbit with their behavior. This "rage" period will usually start to subside within a few days of stopping the med.I have never seen it last more than a week or so after stopping.