Learn Finnish: Pilvet häipyvät, ja aurinko paistaa. - The clouds vanish, and the sun shines.

In Finnish, you must put a comma before ja (and) when it connects two complete clauses that have different subjects. Here, the first subject is pilvet (the clouds) and the second is aurinko (the sun). If the same subject were doing both actions, you would leave the comma out.

Since this lesson is about automative verbs (actions happening by themselves), does häipyä belong to that group?

Yes, in terms of its meaning! While it doesn't have the typical -tua/-tyä suffix, häipyä (to vanish / fade away) describes an action the clouds do entirely on their own. If you wanted to say that someone actively makes the clouds disappear, you would use its transitive counterpart häivyttää (to fade or blur something out).

I've seen paistaa used in cooking contexts. Does it really mean 'to shine' here?

Yes! Paistaa is a very versatile verb. It means 'to shine' when talking about the sun or moon, but it also means 'to fry' or 'to bake' when talking about food. The context always makes it clear whether you are talking about the weather or a frying pan.

Can häipyä be used for people, or just things like clouds?

It is very commonly used for people in spoken Finnish, often meaning 'to leave', 'to slip away', or 'to scram'. Be careful with it, though: the command form Häivy! is an aggressive way to tell someone to 'Get lost!'