window for short time fourier transform
You are talking about symmetry in two contexts :-


Gaussian
Window


I will talk about these two contexts one by one

Gaussian

A symmetric 2-D gaussian (I assume 1-D as in 1-D the question of symmetry does not arise) means that it does not have any directional preference (dictated by equal variances $\sigma_x$ and $\sigma_y$) . Now the job of Gabor Transform is to give us a joint time-frequency distribution (i.e telling us about frequency and phase content of a local image patch)
Now consider the following two cases :-


Symmetric Guassian ($\sigma_x$ = $\sigma_y$)
In this case the gaussian does not have any directional preference.



In the above image, suppose you are centered on the central pixel and consider a patch around it which is indicated in the above image by a black rectangle in the center. Following points are important for this image :


Doing a Gabor Transform over that patch would yield a time-frequency representation, which involves all the pixels in the patch accordingly weighted by the symmetric gaussian.
All the directions at a certain distance from the origin get the same weigthage.


However, if a gaussian oriented at an angle of $135^{\circ}$, then :- 


the white principle diagonal pixels would get almost all the weightage while the other pixels will get little weightage.  
The time-frequency representation will be mainly a time-frequency representation around the principle diagonal.
If the values of $\sigma_x$ and $\sigma_y$ are sufficiently small, the gabor transform will be very very close to the time-frequency representation for the straight line which marks the principle diagonal.


Thus, for the Gaussian function,


Symmetric gaussian is non-directional.
Non-Symmetric gaussian is directionally oriented.
Non-Symmetric gaussian will be useful in cases where the time-frequency analysis of a directionally oriented feature in a patch has to be carried out.


Window

The Window size basically represents the time-resolution (or space-resolution in the case of 2-D signals like images). A narrow window means that the focus on some local patch of the image is high and hence a more faithful time resolution of the Time-frequency analysis will be possible.

However, this is limited by the Heisenberg's Uncertainty principle. Although, Gabor Transform attains the theoretical lower bound for the uncertainty principle, it stays within the bounds of the principle. Thus, changing the window size is akin to changing your time resolution.

A Non-symmetrical window (like $3\times7$) is generally not chosen because generally we do not have a reason or a preference for changing our space-resolution in one direction (x or y) relative to other. In case there is a concern for the same, it can be done. However, I have never till now seen such a situation.