Sliders don't move until play button is pressed, but the values still change.
Your event handlers are throwing errors. That causes the slider to not update properly.
When play is pressed, the correct pitch is played initially, but when the pitch slider is moved to set a new frequency, then paused, and played again, it sounds like it's still at the correct frequency, but an octave higher.
After the first strange "octave change", if it is paused and played once again, it stays at the new unexpected frequency like it should in the first place.
That's a bug:
//PITCH SLIDER//
$(function() {
...
osc.detune.value = $("#cur_pitch").val();
//why are you changing detune here?
...
});