Can foul be called with 0:00?

[referee99]

Boys Varsity game I saw yesterday. Home team up big 45-25. A few seconds remain in first half A1 (home) inbounds in the back court. A2 receives pass and dribbles towards division line, presumably for a last second heave, but never gets a chance as time expires. As time expires B1 goes for steal and there is contact.

Players go to their benches, but PA announcer says there was a 'foul at the buzzer' and A2 will be shooting 1 and 1. He steps to the line and hits both.

My first question is this: can there be a player shooting one-and-one with the lane cleared and no time on the clock here (NFHS rules)?

If there was a foul that occurred before time expired would the officials not be obligated to put the amount of time remaining back on the clock?

If the foul occurred after time expired would the official not be handling a dead ball foul and ignoring or calling an intentional foul?