can a neuron release gaba even if it has an damaged gaba gene?

Hi,
This would depend on how the gene is damaged.  Depending on the damage, it could stop functioning completely or release gaba very slowly.  Yes that is correct.  If it is minor damage it will release gaba but very slowly and therefore not enough.  I hope that helps!

i'm not familiar with the gaba gene in detail, but it probably depends on how the gaba gene is damaged. if the gene is damaged in the coding region of the glutamate decarboxylase, the enzyme could act less efficiently that normal. if it is damaged in a non-coding region, the enzyme could be produced in smaller quantity. another possibility is that one allele is damaged and the other is not. in certain cases, having only one functional copy of a gene can be sufficient, or the enzyme can be functional enough that the cell has enough gaba to operate.

Yes, it can.
Release of a neurotransmitter like gaba (gamma aminobutyric acid) is done by vesicles  If damaged gene can produce a damaged protein with no problem with its important recognition site, then gaba protein can be directed to inside of te vesicles via recognition site on gaba, then by means of exocytosis, vesicles can be release into synapse.
Gaba amount will be same with normal pathway. But due to change or any damage in the structure of gaba protein, it may not show its effect fully, or its effect may increase. When activity of gaba decreased, then you can say, there is not enough gaba to reach threshold. Otherwise, if gaba activity increased then we see there are excess gaba activity although amount of gaba is same.