Do I always need to use the definite article before a given name or are there situations / constructions where I am allowed to leave them out?
Example: "Ele é o João."
Not always, it depends on context.
I wouldn't use the article at the beginning of a sentence if the listener and I had mentioned that person before. - João é um aluno muito aplicado. (but you could say "O João é...") - Elisabete é muito interesseira.
In addition to context, whether one uses the article or not largely depends on where one was brought up. My father was from another state in Brazil and his whole family wouldn't use the article: Vou visitar Maria, Fui à casa de Maria, Este carro é de Álvaro. In Rio de Janeiro it's more common to use the article in such sentences. (Vou visitar a Maria, Fui a casa da Maria, Este carro é do Álvaro.
Generally yes, but it depends on the context. Unfortunately, the contexts do not seem to be the same as the ones Centarus mentioned, so maybe the usage is different in Portugal.
You can sometimes omit it, but only in higher registers, and I can only think right now of two contexts:
If it's someone well known.
Nem Salazar diria tal coisa.
Using the article would also be possible, but it would be more informal and imply more familiarity.
"Cala-te", ordenou Carla.
Carla decidiu então aproximar-se do grupo.