I am new to this game, I have been play for about a week now.
and while I have made level 45, I did not make it in by day 6 [tho, I did get the rewards on day 7]
and while I beat the Professor on or about day 7, I never did get any rewards for it as the beginner's button had vanished by that point.

My suggestion is to not to make the Beginner's Reward time dependent but to make them task dependent, so that you get the rewards when ever you finish that task wether you do it fast or slow.