Broadly there are two potential problems with using a parametric normal-theory test with a Likert DV:2. Can I compare the two total scores for neutral vs. sad films with a paired t-test? Every subject rated all the eight films. Or is there a better test for that?
1) The error distribution won't be genuinely normal with Likert data -> the sampling distribution of the coefficients isn't guaranteed normal in small samples -> your Type 1 error rate and confidence interval coverage might be different than nominal (e.g., True Type 1 error rate greater than 5%, or 95% confidence interval coverage that isn't exactly 95%). But in reality the sampling distribution of the coefficients should be really close to normal with N=40, so this isn't a major worry.
2) Using an ordinal variable in a parametric test is not "permissible" according to the represenationalist theory of measurement, because the results will depend on the (theoretically arbitrary) choice of how you code the variables. See Stevens, 1946. However, you have already assumed your item scores are at least interval when you summed them together; that summing process isn't invariant across coding choices either.
There are a few ways to do this, but I'd ask "Why?" We know different films produce different emotions - what's the goal of this analysis?3. How can I compare all the eight films separately, between each other (i.e. without comparing the two summed variables)?
If sources disagree, so will we (personally I think there are almost always much more important things to worry about than normality, like power and preregistration and measurement error; others disagree). But if you do examine normality, look at how closely the differences between conditions approximate a normal distribution (e.g., via qqplots or histogram). Running a statistical test for normality on Likert data is pointless; Likert data cannot be truly normal, because Likert data is discrete not continuous. The question is about how bad the departure from normality is.4. I read somewhere that I need to test my data for normality, but then other sources say I don’t need to…? Do I indeed have to? And how? At what stage?