"Will you go anywhere tonight?"
Translation:Ти куди-небудь будеш іти сьогодні увечері?
When speaking, no. If I say "Ти куди-небудь будеш іти ввечері?" it implies "сьогодні ввечері". But that sentence would translate to "Will you go anywhere at night?" which has the same situation in English: it implies "tonight" even thought it doesn't stay that it's the same day. So in this particular exercise I'd say it's required.
Wooow, cool! Technically it's quite incorrect grammar, according to standard rules (of course, spoken language doesn't care about that at all). Десь/де-небудь etc. is used only for "де" sentences/questions, and кудись/куди-небудь for "куди". Іти/ідеш in this case refers to "куди?", "where to?" rather than "where", so using десь is incorrect. "Ввечорі" is a non-standard spelling/pronunciation. Нині is perfectly fine, but means "now", so doesn't work because "нині ввечері" sounds like "now tonight".
Again, this is a fine colloquial regional phrase, but I think adding it to Duo would be too much, they try to stick to standard textbook rules, too much colloquialism is confusing for learning.
Is it appropriate to replace "куди-небудь" with "де-небудь", if the question was something like: "Ти де-небудь будеш іти у паркі сьогодні увечері?"
No, it doesn't work like that. "Куди-небудь" is used in the sense of "where to", when there's a direction; and "де-небудь" is used in the sense of "where at", a location.
Also, you added "у парку" which wasn't there :)