"I am busy until the weekend."
Someone correct me if I'm wrong: The subject isn't "weekend," but the implied "I." The full sentence would be 私は週まつまでいそがしいです。私 and such are dropped when they are already understood. It's like in English being asked "When are you busy?" and you reply "The weekend." instead of "I am busy during the weekend."
The weekend isnt the topic or subject (2 different things just wanna point out that). we arent using "ga" because we arent making a statement about the weekend. Its a statement about you and your activity (busy). Time is always complicated in languages. If it helps the "maDe" is sort of acting like a marker, to say when your business ends.
Nope, Kanji + Hiragana writings are a thing in the real world too, for real world reasons. Those reasons involve being able to conjugate words for different reasons (early examples being past tense and the negative versions) without needing a completely different Kanji or adding more pronunciations to each Kanji. However, it may be prudent to learn to base stem form of verbs and such, as there are rules that they follow based on what kind of verb it is (う-verbs vs る-verbs as one example)
Looking up the rules all at once can be pretty daunting though, which is why duolingo introduces the different conjugations over time so you can learn as you go.
Duo is insisting that you use the hiragana for 【まで】. Jisho (https://jisho.org/word/%E8%BF%84) says simply that the meanings that the kanji 【迄】intends to convey are usually written out as kana. I interpret まで and から to be particles, and I've never seen kanji used for those, but I could be wrong!