Sourcing and installing a stainless steel backsplash
Don't bother patching the holes in the drywall.
Backsplashes are usually 304 stainless steel. In the US, you could order that from onlinemetals.com. (No, I don't work for them. I recently replaced my range hood and have the same problem as you.)
You should get a metal shop to cut the metal with a jump shear. It is very unlikely that you can get a nice edge for less money using hand shears or a jigsaw.
I'm not sure about the glue. My old backsplash has some kind of adhesive padding behind it. I'll probably try Liquid Nails LN-203.