Your original screws should still work unless your backsplash is really thick then you’ll need longer ones.
Yes. Be careful, shut the breaker. Measure the depth change and fold the 1/8" spacers as needed to make up the depth change.
These are made of some sort of non-conductive plastic.
I would recommend using an "electrical box extender" for that purpose.
I used this to make my receptacles, outlets and switches level after installing a tile back splash. This should help with the fit of the cover plates, since the cover plate screws into the receptacles.