Question 1: I'm wondering to what extent the repeated measures Anova extends the paired t-test? I came across this definition today.
That being true, then, question 2: in a two-way 2x2 Anova, if the differences of scores is normally distributed, then there's no need for the variables be normally distributed? (such as in a paired test) ?
They do not generalize each other.
The repeated measures ANOVA is a general data analysis technique for longitudinal or panel data where many observations are drawn within individuals or clusters. A random effect, usually an intercept, accounts for the correlated errors so that appropriate estimation and inference can be achieved. The random intercept can be thought of as handling hundreds of unknown covariates common in clusters in an empirical way.
On the other hand, the paired t-test is actually just a t-test, where you exploit the cleverness of the design, having two observations within each cluster, and subtract one from the other, again so that the mean differences are independent and heteroscedastic.