Normal data vs normal residuals

I have the following two questions:

1) one of my variables, say y, is normally distributed (shapiro/lilliefors
tests). However, the residuals of
lmer(y ~ x + (1|Subject))
are not normally distributed. Thus, as far as I understand, I can not use
lmer results. I was thinking about replacing lmer by (robust) rlmer. Do you
agree? Is there a better recommended approach?

2) another variable, say z, is not normally distributed but the residuals of
lmer (z ~ x + (1|Subject))
pass the normality tests. May I accept the results of lmer?

The marginal distribution (unconditional distribution of y in the first
case or z in the second case) is not really relevant at all.

Many statisticians think that statistical testing of residuals for
Normality is a waste of time (because such tests will almost always fail
to reject the null hypothesis for small samples and will almost always
succeed in rejecting for large samples, even when the level of
non-Normality is not a practical problem for estimation or inference:
e.g. see
https://stats.stackexchange.com/questions/2492/is-normality-testing-essentially-useless

If you still conclude that the level of non-Normality is a problem, then
transformation of the response variable or robust estimation via rlmer
are both reasonable strategies. Alternatively you could look for a
statistical framework that allows fat-tailed distributions such as
Student t (e.g. brms).

Kind regards,
CSM

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