Normality requirements across tests I would like to understand the reason why normality is required in many tests.
T-tests: I read that what needs to be normal is the sampling distribution rather than the sample distribution. But since normality of the sampling distribution is inferred through the normality of the sample, hence the requirement (unless the sample is large). Why is it the the sampling distribution must be normal?
ANOVA: this is an omnibus test that compares means among them. is the requirement of normality for the sample distributions due to the fact that, given Therefore  I reckon that means must be good models for their respective samples. This  only happens if the sample is normally distributed, in which case the mean is the centre of gravity of the distribution. Is this the reason for the requirement of normality?
Regression: normality is required for the distribution of the residuals. Is this requirement due to the fact that errors (residuals) must be stochastic and not determined by any other variable (in which case the model would not be a good one)?
I think there must be a common theme for all these normality requirements, but I am missing out on it.
(my stats course was very basic) 
 A: The issue comes down to performing inference (things like hypothesis tests and confidence intervals).
Consider a two-sample t-test (equal variance assumption). If we assume that the samples are random samples from populations with normal distributions, so that the random variables (of which the observations are observed instances) are normally distributed (and those random variables are all mutually independent) then under the null hypothesis the t-statistic will have a t-distribution.
If we were to make quite different assumptions about the distribution, the t-statistic would have a different distribution --- but in those circumstances, we would generally be better off using a different statistic.
In the particular case of t-tests (and ANOVA), the distribution under the null is not especially sensitive to mild deviations from normality (moreso in larger samples), so the distribution of the test statistic may still be approximately right if population distribution(s) from which the samples are drawn are not too far from normal.
It would be perfectly possible to perform ANOVA without making a normality assumption. For example, one could compute the ANOVA F-statistic, but without the assumption of normality it doesn't have an F-distribution. If we assume - when the null is true - that the samples are drawn from the same population-distribution, then we could use (say) a permutation test based on that F-statistic; we would have a valid test that did not make the assumption of normality.
On the other hand, we might assume that the conditional* distributions were Gamma with common shape parameter, and that under the null the means are identical. In that case we would have a similar analysis but would not use the usual F-statistic. [Indeed, the usual analysis would be to fit a GLM and use asymptotic results related to likelihood ratio tests.]
* conditional on the value of the variable indicating group membership.

Therefore I reckon that means must be good models for their respective samples. This only happens if the sample is normally distributed, in which case the mean is the centre of gravity of the distribution

The mean is the 'center of gravity' whether or not you have normality. Means can be useful/representative when the data are not drawn from normal distributions. 
