how accurate frequency does a PC speaker produce
The classic PC Speaker is a very simple system that was designed to be inexpensive and use off-the-shelf parts of the early 1980s.

The core problem is that the PC speaker system consists of a cheap speaker, a timer, and digital pulses. Using this system, you can get quite a lot of interesting sounds out that weren't originally intended. But you are limited to square waves and the harmonic content that comes with them.


  I don't think there is any spec I can read about frequency response.


The best you can find is specs for the timer and then run experiments with the actual hardware. The early IBM PCs had discrete timers, but at some point that functionality got rolled into the VLSI chipsets that PC makers used to build motherboards.

If you have an actual audio chip built into the motherboard (fairly common, these days, much less common even in the late 1990s), then it is likely that the PC speaker functionality is emulated there. So the behavior will be different from the original hardware.

Your integrated sound card should produce the most accurate sound. But at the upper end of the sound range, you may be limited by the physical capabilities of your laptop speakers. If you drive the speaker faster than it can move, then you won't get the sound you expect.