So, if the clock in the gym on the wall, goes to the hundredths and shows 0.39 then can't catch and shoot?

When (enough of) those clocks are used, then the ruling bodies will adjust the rules.

Also, on the clock issue above -- most game clocks work as I described. Most shot clocks work as jeremy341a described.

Who’s Trent Tucker? And Why Is There A Rule Named After Him? ...
 

Here's my rewrite.

The “Trent Tucker Rule” disallows any regular shot to be taken on the court if the ball is put into play with three-tenths of a second or less left in the period. The rule was born out of a game between the New York Knicks and the Chicago Bulls on January 15, 1990, at Madison Square Garden. The game was tied at 106 with one-tenth of a second left in regulation and the Knicks in possession. During a timeout called by the Knicks, both teams prepared for what was seen as the only possible way the Knicks could win in regulation, an alley-oop tapin by Patrick Ewing from an out of bounds pass. When play resumed, the inbounding Knicks player, Mark Jackson, saw the alley-oop play get broken up. He proceeded to throw the ball inbounds to Trent Tucker (Minnesota 1978–1982, NBA 1982–1993), who was the only Knicks player open. Tucker then turned around and hit a three-point jump shot before the buzzer, giving the Knicks the win, 109–106. Replays showed that the clock had not started until Tucker's shot was already in flight. Afterward, it was determined that a player could not catch, and release, a shot that quickly, and after the rule change, all an inbounder could do, with three-tenths of a second or less left in the period, was to pass the ball to a teammate and have them tip it in.

In addition to Trent Tucker, my article will detail rule changes associated with Leroy Edwards, George Mikan, Bob Kurland, Bill Russell, Wilt Chamberlain, Lew Alcindor (Kareem Abdul Jabbar), Ralph Sampson, Darryl Dawkins, Shaquille O'Neal, Patrick Ewing, Karl Malone, Hakeem Olajuwon, Allen Iverson, and Skylar Diggins. Most of these rule changes were originally instituted in NCAA, or NBA games, but these changes eventually trickled down to NFHS rules.