Deep as in ‘How big a spacer does the hem become?’; or ‘How wide is it?’
The panel is 24 ga., so it’s very thin. Caulk or mastic would probably be thicker. Its width is 1/4" or 1/2". I only noted it was there, and didn’t measure it.
Why would it warp? It’s being applied to a flat surface. It won’t be exposed to heat, as there is sufficient space between the range and the wall. Though it’s called a ‘backsplash’, it’s unlikely that anything will splash on it. The drywall has survived. The worst it will see is greasy smoke when I use the cast iron.
FWIW: I’ve mentioned this is a very old house. Dimensions tend to be non-standard. Eventually the DIY cabinetry someone installed will be replaced by cabinetry that is as standard dimensionally as possible. Thus, the backsplash may be repositioned or replaced when the time comes. The space for the stove is 36". The stove is 30". The backsplash will be placed such that it covers the 36" space. At 30", it is tall enough to extend below the top of the stove. When they installed the stove, they left an inch to the left (next to the counter), so there is a 5" space between the stove and the refrigerator. When we get new counters, I plan to have a bit of counter space added between the stove and the fridge. This will fill in the empty space, ‘hide’ that the backsplash does not go to the floor, and provide a handy place to put cooking utensils when they are in use as well as preserve the ‘handle room’.
But cabinets and counters (and a new ceramic tile floor) are expensive, so this will be a project long in progress.
I’ve seen instructions online. Some say to use construction adhesive applied with a caulking gun and spread with a putty knife. Others say to use mastic and a toothed trowel. It appears that mastic spreads more easily, and it is easier to achieve an even application. It also seems easier to apply mastic to a wall, than to apply adhesive to the sheet metal. OTOH, I’ve never used mastic.