What are main reasons a typical DSGE model is called non-linear? Why are these non-linearities important? Is a non-linear model inherently less stable than a linear model? Why?
They are called non-linear because they often consist of non linear equations (although linearized approximations exist).
No matter what kind of empirical research you are doing it is important to avoid misspecification. If relationships are non-linear it’s not appropriate to try to fit an linear model which identifying assumption requires linearity. If you would do so your parameter estimates would be biased.
There is nothing inherently unstable about non linear processes. In the context of DSGE models dynamic stability implies there should be no unit root in the modeled series. There is no reason why non-linear series would be more prone to have unit root than linear one.