As AnnaMarie T. said, "He turned twenty-two (today/yesterday/last week)" is correct. "He'll be turning twenty-two next month." If the birthday is in the near future, you can also use present tense: "I turn twenty-two next week" or "I'm turning twenty-two next week." During a particular year, we usually just say "He's twenty-two." It's also possible to say "He's in his twenty-second year," but this is rather formal and not very common.

Yes, you say, “ He turned 22 (today).” I’m not sure I understand your second question. “He’s 22” means he’s still that age (until he turns 23).

Thank you, Gary!

Just "He is 22 today" is also fine.

If you don't know his age or you don't want to mention it, you just say, "It's his birthday, today", or "It'll be his birthday, tomorrow" (if it is tomorrow). Otherwise, you just mention the age he will be. ie "He's 22 today" or "He'll be 22 tomorrow".