If the von mises stress well is below the yield stress, am I right to assure the material is behaving elastically?
Possibly, but only for a single or low number of cycles. If you're dealing with cyclic or repeatedly-applied loads, you'll need to account for the material's fatigue strength.
It depends on material. Von-mises yield criterion is good for steel and similar ductile and homogenous materials. For concrete, ceramics etc it is not accurate because of the different behaviour under tension and compression, where other yield criteria can be used like the Drucker–Prager one. For soil the Mohr-Coulomb criterion is often used.