Learn French: Oui, j'ai un rhume et un peu de fièvre. - Yes, I have a cold and a little fever.

The word for fever is indeed feminine (la fièvre), but here it follows the expression un peu de (a little bit of). After phrases that describe a quantity—like un peu de or beaucoup de (a lot of)—the word de locks into place and does not change to de la, du, or des.

Is un rhume the same word used when you feel cold because of the weather?

No, un rhume strictly means the medical illness "a cold". If you want to say that your body feels cold because of the temperature, you would use the word froid and say j'ai froid (literally: "I have cold").

How does this sentence fit into a lesson about pourquoi (why) and parce que (because)?

This sentence is a very natural answer to a pourquoi question, such as "Why aren't you at work today?". While you could start your reply with Parce que j'ai un rhume... (Because I have a cold...), French speakers often drop the parce que in casual conversation and just state the reason, exactly like we do in English.

Do we use the verb avoir (to have) for all sicknesses here?

Yes! Just like in English, you use avoir to say you "have" an illness. You say j'ai un rhume (I have a cold) and j'ai de la fièvre (I have a fever). This is one of the many times French and English match up perfectly.