ADS1115 PGA Questions
The board will not fry if all analogue input voltages are within the range 0.3v below ground to 0.3v above Vdd, regardless of the PGA setting.

If the measurement voltage is outside the maximum voltage range according to the PGA setting, the digital code will be clipped at the maximum (or minimum).

If the differential input voltage is 0.2v, regardless of the common mode, which can be anywhere from gnd to vdd, fir example your 3.8 and 4.0v, with a vdd >= 4v, the most sensitive range for measurement will be gain 16, +/- 256mV full scale.