I always thought both was 'både' am I wrong or are there multiple forms? Whats the difference?
Yes, there is både and båda.
Både is a conjunction, meaning it links things together. That's why you'll say Jag ser både huset och trädet ("I see both the house and the tree").
Båda is a pronoun, meaning it refers to the both of something. That's why you'll use båda in the sentence here.
Does that make it a bit clearer?
båda och is not used. både is always used together with och, and sometimes you can have både och as a standalone expression. I wrote a little more about it here: https://www.duolingo.com/comment/8387413$comment_id=15263308
Would "I know both of them" be an acceptable translation for this sentence? "I know both" seems a bit clumsy in English (not wrong, just clumsy).
The audio for this one seems completely off. I listened too it ten times but could not figure out that she was saying "Jag".
So bada is when "bade" is the subject, but bade is when you're using it to connect words?
EX: Jag ser bade husen och kvinnan.
Doesn't work in that specific context. båda would have to refer to a noun, like 'both of them', but 'happy' and 'sad' are adjectives. Känner du dig glad eller ledsen? Jag känner mig både och is ok.
No. It works for "feel" as in "sensing with touch", like you walk into a dark room and are asked if you can feel the chair and the table.
"feel" as in "having an emotion" is reflexive, "jag känner mig xx", "känner du dig xx?" and I can't think of a way to use it with "båda". One would probably answer "både och".