"No, thank you."
"Dziękuję" is quite neutral, "dzięki" is informal. So you will rather use "dzięki" with friends only, or at least people you feel comfortable with. "Dziękuję" will sound ok even with them.
"Nie, dziękuję ci"? Really? Either "Dziękuję, nie" or "Nie, dziękuję" and if someone must add "ci" then I guess it must be written "Ci" as all words like: Ty, Twój, Ci, Wam, Wy, Wasz start with capital leter. Just like "I" in english (as oposed to "ja" in polish)
The basic translation is "Nie, dziękuję", "Nie, dziękuję ci" is just among the accepted options.
I also think that all such words should be capitalized, yes. However, it happens when you address someone directly. So sure, if I was writing this answer in Polish, I would write "Tak, zgadzam się z Tobą".
However, the Duolingo exercises are just sentences without any context. They are not addressed at anyone. So a small letter is used. Just like, for example, if you were reading dialogue in a book.