"Are you busy on weekdays?"
In this case you cannot substitute に for は. Weekdays becomes the topic and you use the XはYです sequence. 'As for weekdays, (I am) busy. There are times when you can use に, but not in this structure. This is because of the involvement of いそがしい. As this is an adjective, you must use は. Even a specific day of the week would follow this pattern (水曜日は忙しいですか？）’The work week is long would be 平日は長い'。This sentence would be better translated (literally) as 'Are weekdays busy?'.
は after 平日 is optional in this case though. I wrote へいじついそがしいですか？ and it was accepted. I think in this case the は is just for emphasis and the "I" (or whatever pronoun the sentence happens to be about) is omitted as usual.
I wrote "平日は忙しいか" and for some reason it was wrong because I put "か" instead of a question mark. Why does this matter?
You left out です. Duolingo likely requires this 'proper' phrasing.
Technically, question marks should be optional.
Not proper so much as polite. In casual conversation, omitting the copula (x is y) is totally acceptable. For some reason Duolingo only teaches polite form, like you said.
Shouldn't 平日 use が instead of は? "You" is the topic, weekdays are the subject. At least write "平日はあなたがいそがしいですか。" When there's no mention of someone else, it's assumed you're talking about yourself, or something in general (for example, "are weekdays busy?" is what I originally thought when I saw this). I know this is somewhat nullified by か but it still exists.