Learn Swedish: Kvinnan klappar hunden. - The woman pats the dog.

It means both! In Swedish, klappar is the standard verb used for petting an animal, as well as giving a quick pat. You can also use it for clapping your hands, as in klappar händerna.

Why is it hunden and not den hund for "the dog"?

In Swedish, the definite article "the" is attached to the end of the noun when there is no adjective describing it. You only use the freestanding word den when there is an adjective involved, such as in den snälla hunden (the nice dog). Since there is no adjective here, it is simply hunden.

How would you say "The woman is petting the dog" instead of "pats"?

You would use this exact same sentence. Swedish does not have a separate present continuous tense like the English "is petting" or "is patting". The present tense klappar covers both "pats" and "is petting" depending on the context.