Learn Spanish (Spain): ¿Te importaría hacer un flan para la cena?

Using the conditional tense (importaría) makes the request much more polite. Instead of asking "Do you mind?" (¿Te importa?), you are asking "Would you mind?" (¿Te importaría?).

How does the verb importar work in this sentence?

Importar works just like the verb gustar. The literal translation of this sentence is "Would making a flan matter to you?". The pronoun te means "to you", and the action of making the flan (hacer un flan) is actually the subject of the sentence.

Why is hacer left in the infinitive instead of being conjugated for "you"?

Because the person you are speaking to is the one making the flan, you just use the infinitive (hacer). If you wanted them to let someone else make it, you would have to use que plus a new conjugated verb.

Why do we use para instead of por before la cena?

We use para here because it tells us the purpose or destination of the flan. It is being made specifically for the event of dinner.