Is it customary to correct for the gain of a window?
Yes, it is customary to correct for the gain of a window, except for some cases I refer to later. (If you are interested only in the relative amplitude, of course you do not need to correct for the gain.) 

Because the window reduces the gain of the original signal (time domain), the amplitude obtained through FFT need to be corrected. For example, if you use the Hanning window, you need to multiply all the amplitudes by 2 (the reciprocal of 0.5). As I understand it, most of the software packages for FFT automatically correct for the window used.

However, such correction is good only when all the frequencies of interest distribute throughout the time domain window. For example, suppose you have 1024 data with all signal levels of zero except for #512 point which has a value of 1 (impulse signal). Obviously, any windows do nothing to the data. So, if you correct the amplitudes for the window gain (multiply by 2), then you will end up with overestimation of the amplitude. If your 1024 data are all zero except for the very 1st point with a value of 1, then every point has a value of zero after windowing, and you lose the signal.

So, if you are dealing with random signals, with all the frequency components expected to lie almost evenly over the length of the signal, you need (or should) correct for the gain of the window you use.