Low carb and low calorie for a month sounds rough in general. Maybe the depression could be attributed to exhaustion and a suppressed metabolism as well.
Very individualized. I have drastically better gut health, skin health, energy when on a low carb diet for an extended duration.
Maintaining that diet for an extended duration, however, is always the challenge when others in the household do not follow it. Logically, I should stop being a baby and just power on with it. Or at least determine what in my "normal" western diet negatively impacts gut health/skin health and overall energy.