I am confused as to when the start of the day for my Duolingo day is. I am unable to have successful streaks because my Duolingo days start and end at normal times.
In your Home page, to the right, under your Streak count, there is a number that says: 'x hours left'. That is what tells you when your Duo day starts/ends.
Duo doesn't provide a way but there was a post on these forums recently which mentioned a user script that could change it although I have never tried it myself. Unfortunately I cannot remember the URL of the post, perhaps someone else bookmarked it and can let you know.