You could say, "why does he have a beard", but "why has he grown a beard" would be preferable. Native English speakers would NOT say "is he having". You can say, "why is he growing a beard".

“Why is he growing a beard? “ is correct, but the other sentence isn’t. You should say “why does he having a beard?” instead.

I would never say "Why is he having a beard?" That sounds weird. I would say "Why does he have a beard?" (if he already has one) or "Why is he growing a beard?" if he's in the process of growing one.

The only reason a native English speaker would say "Why is he having a beard?" is if we were watching some weird sci-fi movie and the character in question had been impregnated by aliens and was giving birth to a bundle of hair in the shape of a beard. In that moment, someone might ask their fellow movie goers "Why is he having a beard????" That almost never happens. I would totally watch that movie. If you saw an actual human person with a beard, the questions you probably want are "Why does he have a beard?" or "Why is he growing a beard?" It does not matter when you see them nor how much time has passed since you last saw them. However, that's kind of a rude question. You probably shouldn't ask it. But if you absolutely must know, "Why is he growing a beard?" is the correct way to be nosy.

You can use both sentences. That's no problem! :)