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NFHS chose 35 for 2 reasons, because they wanted to be different from college basketball, and because college rule changes take time to trickle down to NFHS. They went with the old college values.
Patting the head was an informal end-of-period signal, so when the shot clock came in existence, the rules makers chose to give it formal status as the shot clock singal. Note: in FIBA, officials tap the shoulder instead to signal a shot clock violation.
The secondary value was derived, at least for the 24 and 30 second shot clocks, with a subtraction from the full value to account for the 10-second count. Back when it was a 35/15 shot clock, 15 was somewhat arbitrary. However, later it became more systematic. I suspect NFHS will use 25 as the secondary reset once the 10-second count gets pegged to the shot clock.
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