Lets consider a pure sine signal at $\nu$ that is chopped using square pulses (like a burst mode on signal generators). My understanding is that instantaneous frequency is $\nu$ when oscillations are ON and 0 when they are OFF. On the other hand fourier spectrum is constant over time and contains also other frequencies, since it is not pure sine anymore. Is this correct? which one is used when calculating some frequency dependent physical quantity?
Yes, your understanding ist correct. Instantaneous frequency is the time derivative of the sine argument. As Robert mentions in his answer, this argument is not defined where there is no sine (or complex exponential) function but I think its reasonable to consider it a sine with amplitude zero and constant argument. The function you describe is defined sectionwise. In sections where the sine is "on" the time derivative of its angle is $\nu$, in sections where the sine is "off" the time derivative is zero. So the instantaneous frequency is a function of time.
The Fourier transform is not the right tool to analyze the instantaneous frequency as a function of time. As you have realized the Fourier transform is constant in time. The FT of this special function is a shifted sinc function and thus contains other frequencies than $\nu$.
Update following your comment: 2 is correct. The output signal of a narrow bandpass filter with center frequency $\nu$ is not identical to the discussed "chopped" sine wave. The input signal has sharp transitions where it is forced to zero by the rectangular pulse train. These transitions are smoothed by the bandpass filter and you will see the dynamic behaviour of the filter in form of transients in the output signal where the input signal has sharp transitions. In other words: the bandpass filter can not "react" instantaneously to the sudden change of frequency because it has a memory.
the frequency of a sinusoid is normally thought to be undefined when there is no sinusoid (i.e. when it is turned "OFF").
now, there are at least a couple of different methods of turning it OFF. the most obvious is to effectively multiply the waveform output by zero to disconnect the sinusoid from the output.
however if the output of your signal generator is always connected (so there is no multiplication by zero) to the look-up table (LUT) that has the waveform values in it (and let's say that those waveform values follow a sinusoidal function), then another way to stop the output is to stop advancing position in the LUT. that would be like setting the instantaneous frequency to zero. but, unless you're watching out for this, it could get stuck on a DC value that is non-zero. so there are a lot of functions, besides the zero function, that have a frequency of zero. any DC function has instantaneous frequency of zero.