If you do quests from lower ranks you will get less progress. It doesn't speed the whole thing up, it slows it down.
Instead of doing 4 tribes, 3 quests each, you simply do 3 tribes. Not that hard. When you rank up and are below 7 you can take 3 more quests on the same day.
Knowing when daily reset is doesn't speed things up unless you happen to have spent less than 2 when reset hits. It's the normal daily reset time. You can look at it at "timer".