Learn French: Est-ce que c'est parce que tu as un rhume ?

No, because pourquoi means "why" (asking for a reason), and parce que means "because" (giving the reason). Even though this sentence is a question, you are guessing the reason yourself ("because you have a cold"), so you use parce que.

Why does que stay as a full word instead of shrinking to qu'?

You only shorten que to qu' when the next word starts with a vowel or a silent 'h' (like in parce qu'il or parce qu'elle). Since the next word here is tu, which starts with a consonant, parce que stays as two full words.

What does Est-ce que c'est mean word-for-word? It looks very long for just "Is it".

Literally, it means "Is it that it is". Est-ce que is the standard French phrase used to turn any statement into a yes/no question. The actual "it is" part is c'est. Put them together, and you simply get "Is it?".

Does avoir un rhume work just like "to have a cold" in English?

Yes, it is an exact match! You use the verb avoir (to have) for catching a cold. This is nice and easy, especially since French sometimes uses avoir where English uses "to be" (like tu as faim for "you are hungry").