logical operators fall below increment operations in the order of
precedence.
Order of precedence is not order of execution. They’re completely different concepts. Order of precedence only affects order of execution to the extent that operands are evaluated before their operator, and order of precedence helps tell you what the operands are of each operator.
Short-circuiting operators are a partial exception even to the rule that operands are evaluated before the operator, since they evaluate the LHS, then the operator has its say whether or not to evaluate the RHS, maybe the RHS is evaluated, then the result of the operator is computed.
Do not think of higher-precedence operations “executing first”. Think of them “binding tighter”. ++ has higher precedence than &&, and in the expression x ++ && y ++, operator precedence means that the ++ “binds more tightly” to y than && does, and so the expression overall is equivalent to (x++) && (y++), not (x++ && y) ++.