hyper elastic is used for flest and bone or rubber like material. don't use it for metal
here is the difference between linear and non linear in the case of metal:
For linear study (Hook's law) , stress = Young Mod * strain.
1 Truth stress strain curve
2 elastoplastic with perfect plastic deformation
3 elastoplastic with strength hardening.
Most of engineering application (95%) falls into the linear elastic.
If the secondary stress (caused by contact or geometry change) is too high, None linear can be employed. Don't look at the stress level. examine the equivalent plastic strain (guild line per API-8C). it was good to study the affect of the local singularity of the geometry.
Answer to your Q1. Stress in None linear study does not follow the Hook's law, and thus it is smaller than the other
Answer to your Q2. Both are good.