How to prevent fitted bottom sheet from slipping off mattress?

Sew elastic at the corner's so it will hold the sheet under the mattress.

These straps are the perfect answer and keep them snug and smooth: https://amzn.to/2RvkQKv

Is it too large or too small, as regards depth of mattress vs. depth of sheet?

A sheet designed for a shallower mattress can be made to fit by cutting 2" elastic to fit the corners. Using pinking shears, cut 45 degree opposing angles for each of the 4 pieces of elastic (your elastic should be tight enough to stay on the mattress, but not so tight that there is no " give".

Attach the elastic strips with safety pins, then sew in place using your machine. It's best to make a small hem at either edge of each corner to increase longevity.

Wash and dry sheets on no more than warm wash, low heat dry.

You'll find videos of this on You Tube.