The lottery quest will reward you with a free lotto ticket, not a loot chest, so that's by design (and naturally won't appear in the chest history as it is not a chest). Support can double-check your Quest rewards for you if you submit a ticket to them.
If you have referred someone you can collect your rewards from your Profile. You should see a "Referrals" tab, click Tell Me More and it will show you any active people you have referred and this is where you would be able to collect your rewards for those referrals.
I have that on screen however all the collect boxes are locked. Week one and three are full points needed.
I’ve been trying, they have been locked ever since he joined three and half weeks ago, last week was week three and it’s locked like the rest??