confusion about the link between residuals, error terms, sample size and CLT in ANOVA

I feel a little confused about the assumption of the ANOVA and what it ensures mathematically

1. the errors have to be iid and normally distributed N(0,1).
2. independance of observation. Is it not a consequence of the first assumption?
3. homogeneity of the variance inside each group.
4. the residuals have to be normal.

I have read that we cannot make a link between the residuals and the true errors. Indeed, we need the error to be assumed normal. Then what is the point when checking residual's normality if it not to make an hpothesis about errors' distribution?

About the sample size (I assume we talk about the overall number of individuals, correct me if I am wrong) :

• does a big sample size ensure that the residual normality can be assumed with the CLT? or
• does a big sample size ensure that we will be able to approximate the F distribution by the Chi square distribution. What is the link between the sample size and the residual or the CLT in ANOVA? What will be the consequence (do the p-value be more trustworthy?)

Maybe I have all mix it up. If someone can clarify the statements above, it will help me a lot. Do not hesitate to give formal proofs or references.

Thanks again.