What is model adequacy checking in anova?

#1
In anova, we assume that the random error term \(\epsilon_{ij}\) is independent and identically distributed with mean \(0\) and variance \(\sigma^2\)

How can I check the normality assumption and independence assumption?

Does model adequacy checking mean "Checking the normality assumption that \(\epsilon_{ij}\sim N(0,\sigma^2)\)"?More specifically checking the residual, \(e_{ij}=y_{ij}-\hat y_{ij}\)?And is independence check performed by showing `Mean square due to treatment` and `Mean square due to error` independently follow chi-square distribution according to Cochran theorem ?