I believe the timer-counter is derived from the main CPU clock via the prescaler, so changing the timer-counter prescaler should not affect the main clock.
Since the delay functions are based on counting clock cycles, they should remain tied to F_CPU. (So long as F_CPU is defined correctly in your boards.txt file and you've burned the fuses accordingly.)
However, the millis and micros functions are based on counting timer-counter overflows, and this will be affected by changing the prescaler.
Also note that there are two PWM modes, fast and phase correct. The core files set fast mode by default. If you switch to phase-correct [TCCR0A = _BV(WGM00);] the frequency is halved. (Actually 256/510.)
Actual output frequencies will not be exactly as listed in the ATtiny13a manual. 10% slower is not uncommon. If you need an exact frequency output, you'll need an external clock.