Generating very short signals
If you want to generate a very short pulse, it is not usually a problem to have it at any time you wish to have it. That is because modern audio interfaces (sound cards) have an interpolating lowpass filter in the output path, usually with an impulse response that without latency looks like the sinc function:

Figure 1. Sinc function, shown truncated at the ends and critically sampled with the sample number shown in the horizontal axis.

If you play back audio data that is sampled from a time-shifted sinc function with the peak somewhere between sample points, the interpolating lowpass filter will reconstruct the function from the samples so that the peak is in the correct position. Some audio interfaces use a low-latency variant of the lowpass filter that does not have a symmetrical sinc impulse response. Disregarding latency, it has just as good inter-sample timing capabilities if you give it samples of a time-shifted sinc function.

If you need a more rectangular pulse, you could add an analog comparator to the output of the audio interface. You can change the pulse length by varying the voltage level you compare to.

First test if you get that sinc shape from your hardware, like MBaz suggested.