Learn French: Est-ce que tu tousses toujours ?

You are right that toujours often means "always". However, it is also the standard French word for "still". You can usually figure out which meaning is intended from the context. Here, "Are you still coughing?" makes more sense than "Are you always coughing?".

Why does the sentence use tu tousses instead of translating "are" and "coughing" separately?

French doesn't have an equivalent to the English "am/is/are doing" form. The simple present tense tu tousses translates to both "you cough" and "you are coughing".

What is the purpose of Est-ce que at the start of the sentence?

Est-ce que is a standard phrase used to turn a regular statement into a yes/no question. The statement here is tu tousses toujours (you are still coughing). Adding Est-ce que at the very beginning flips it into a question.

Is the -es at the end of tousses pronounced?

No, it is silent. For regular -er verbs like tousser (to cough), the -es ending used with tu makes no sound. You just pronounce the base of the word, which sounds like "toos".