You do not have a nonlinear model, you have a linear model with a polynomial predictor. So yes, this model is nonlinear in the predictor
x, but it is linear in the unknown model parameters, and that is what is meant with a linear model.
So your model assumes that, around the parabolic expectation, there is a normal-distributed error term with zero expectation and constant variance.
So to your question:
My question is, to test the validity of the model, is testing for
autocorrelation, heteroscedasticity and normal distribution of
residuals relevant for non-linear models?
Yes, yes, and yeas, all of those are potentially relevant for your linear model. And, if you had a nonlinear model with additive normal errors, those would still be relevant.