how to adhere a stainless steel backsplash to painted drywall surface
I would assume that it'll get warm at times and use a product that won't goo apart as a result. Anything that's stiff will leave you with ripples visible from the side. 

A panel adhesive that lists suitability for metal will give you a smooth finish and lasting bond. Use a small v-notch trowel to distribute it flatly and evenly to avoid rippling.

Alternatively, attach a sheet of felt to the back with double-sided tape to act as a vibration and noise dampener and just use the four screws.