Japanese word not Japanese?
Please correct me if I'm wrong.
It's my understanding that katakana is used for words that Japanese took from other languages. On the Hobby 3 skill, I'm given the word 'karaoke' but it's in katakana. That tells me that the word does not originate from Japanese, but if I look it up online everything says that it did. I'm not so much asking the origin of the word, but I'd like to know if there are exceptions to the 'katakana=not of Japanese origin' rule.
I might be wrong, but I don't think Katakana is strictly for words that originate from other languages. A quick search will confirm that Karaoke is a Japanese word, it means "Empty Orchestra." To my understanding, Katakana share sounds with Hiragana as well, so it is Japanese.
As you mentioned, 「カラオケ」 has been constructed from 「空」 (read 「から」) and 「オーケストラ」. So, one of the "building blocks" of the new word, 「オーケストラ」, already seems to be a loan word and therefore written in Katakana. I'd assume that because of that the new word is also written in Katakana, even though the term originated in Japan.
As TheEeveelord pointed out below, even if the word is 100% Japanese, It will be written in Katakana for certain purposes, some scientific, others just because it's "cool" and "hip"
This is a common misconception about katakana. Not all katakana words are loan words. Katakana is used for some words in Japanese, a lot of onomatopoeia is written in katakana, and used for the dialogue of robotic characters. Katakana is also used to emphasize words, especially in children's books. In addition, katakana is sometimes used to show yelling.
There are plenty of natively Japanese words that are written in katakana. Lots of otomotepia is written in katakana, like ドキドキ, which is the sound of heartbeats (you can actually combine it with します to mean something like "my heart is beating fast").
I've read that the names of animals are often written in katakana in scientific contexts, so a scientific study would probably write "cat" as ネコ rather than 猫 or ねこ.
Some words are writen in katakana for no real reason other than spelling convention. For example, バカ (idiot, fool) and ラーメン (ramen).
You summed up nicely some of the the main cases where katakana is used that I have very little to add here: sometimes katakana is used to convey a different impression from that conveyed using kanji or hiragana.
Example: if the katakana word (センセイ) is used instead of the kanji 先生（せんせい = sensei）to talk about or address a teacher, it will imply less connection with (or depending on the context, less acknowledgement of) the job (or status) of being a teacher.
A parallel example: the use of foreign words themselves to convey a different impression, like the use of the word ローン (loan) to sooth down the otherwise heavy feeling that would accompany the original Japanese expression written in kanji 借金 (しゃっきん = shakkin) that also means "loan".
P.S. ラーメン too is a loan word as it comes from China. In addition, onyomi readings of kanji are also written in katakana.