When asking for permission, “may” was traditionally viewed as correct and “can” as incorrect. Nowadays “can” is widely accepted in that usage and predominates in everyday English. Where “may” is still used, it is more common in the first person (“May I/we?”). Third-person uses like “May the young man have a try?” and “May he go home now?” risk sounding quaint, like something out of an old book.
“Shall I?” has less of a feel of asking permission and more of making a suggestion. “Shall we?” is making a suggestion. In modern English this pattern is hardly used in third person, and “Shall the young man have a try?” and “Shall he go home now?” do not sound very natural. If you want to make a suggestion then say it another way, such as “Why don’t we let the young man have a try?”