How would I use data to find specific humidity and mixing ratio?
The online calculator is giving you the specific humidity, which at a relative humidity of 43% is 43% of the saturation specific humidity at that temperature. 

Reading from the chart, a rough estimate for the saturation specific humidity is ~37.5 g/kg at 35°C. 43% of this is 16.1 g/kg, which is close to the value given by the calculator.