Is the assumption of normality of the error term needed to use p-value? I have been thinking lately about the following: 
1. Is the normality assumption of the error term really needed in order to make use of p-values for linear regression models?
A previous CV post (see also image below) showed that the estimator of a linear regression model is normally distributed, given that the error is normally distributed. 
However, the normality assumption of the error term is not really needed in order to make use of p-values given that we have a large sample size $(N>30)$, right? The derivation of the asymptotic normality $\sqrt{T}(\hat{\beta_T}-{\beta}) \rightarrow N(0,\sigma^2)$ did not make use of the normality assumption of the error term.

Secondly, I have read many posts regarding the use of p-values in the case of robust linear regression models. However, I do not see why the use of p-values is complicated when we make use of robust linear regression. 
2. What complicates the use of p-values in the case of robust linear regression models? And how would one assess the significance of the coefficients in a robust linear regression model?
EDIT: Derivation of asymptotic normality of estimator of linear regression only makes use of the mean property:

 A: I'm not a statistician, but I'll try to weigh-in.
Note: edited somewhat in response to comments by OP, to make this response more relevant for the linear regression case.


*

*For the p value to be valid, the assumptions for the test need to be met.  These may include normality of errors for some tests, but other tests will have different assumptions.  That is, if alternatives to classic linear regression are considered, the assumptions of p values for parameters for those models may not include that assumption. Questions about what assumptions or complications go into considering the reliability of robust models depend upon the specific robust model being considered.

*Relying on the central limit theorem to ensure the normality of parameters without examining the data is not a great practice. When thinking about e.g. if the sample mean is likely to come from a normal distribution, if the population distribution is very skewed, it may take a very large sample size to ensure an approximately normal distribution of means (and 30 is not necessarily a large sample size). (See Rand Wilcox quote below which considers specifically the two sample test, but I think is instructive.)  It's not entirely clear to me how these considerations can be used practically in assessing the appropriateness of classic linear regression.
Rand Wilcox, 2017, Modern Statistics for the Social and Behavioral Sciences.  Section 7.3.4.

Three Modern Insights Regarding Methods for Comparing Means
There have been three modern insights regarding methods for comparing
  means, each of which has already been described. But these insights
  are of such fundamental importance that it is worth summarizing them
  here.
• Resorting to the central limit theorem in order to justify the
  normality assumption can be highly unsatisfactory when working with
  means. Under general conditions, hundreds of observations might be
  needed to get reasonably accurate confidence intervals and good
  control over the probability of a Type I error. Or in the context of
  Tukey's three-decision rule, hundreds of observations might be needed
  to be reasonably certain which group has the largest mean. When using
  Student's T, rather than Welch's test, concerns arise regardless of
  how large the sample sizes might be.
• Practical concerns about heteroscedasticity (unequal variances) have
  been found to much more serious than once thought. All indications are
  that it is generally better to use a method that allows unequal
  variances.
• When comparing means, power can be very low relative to other
  methods that might be used. Both differences in skewness and outliers
  can result in relatively low power. Even if no outliers are found,
  differences in skewness might create practical problems. Certainly
  there are exceptions. But all indications are that it is prudent not
  to assume that these concerns can be ignored.
Despite the negative features just listed, there is one positive
  feature of Student's T is worth stressing. If the groups being
  compared do not differ in any manner, meaning that they have identical
  distributions, so in particular the groups have equal means, equal
  variances, and the same amount of skewness, Student's T appears to
  control the probability of a Type I error reasonably well under
  nonnormality. That is, when Student's T rejects, it is reasonable to
  conclude that the groups differ in some manner, but the nature of the
  difference, or the main reason Student's T rejected, is unclear. Also
  note that from the point of view of Tukey's three-decision rule,
  testing and rejecting the hypothesis of identical distributions is not
  very interesting.

A: 1)
No. I recall normality condition is needed only for small sample (finite sample) cases. For large samples, asymptotic normality holds given some assumptions.
(See Halbert White, Asymptotic theory for econometricians. Sorry for non-Statistic biased recommendation)
"Large" is ambiguous, but often say something like N>30. I suspect this came from t-dist and normal dist, that t-dist(N>30) becomes close enough to normal dist. Maybe you will find about this in Statistics. At least I haven't heard of Econometrics, though.
2)
Though I don't have a clear answer for this, I guess this is coming from robust estimators' distributions being less trivial with finite samples. In this case, bootstrapped p-value can be used instead. One resamples, estimate coefficient in order to get "empirical distribution of coefficient". Then p-value can be calculated.
(reference: Wooldridge, Introductory Econometrics: A Modern Approach, Econometric analysis of cross section and panel data - sorry again for the "tilt")
