For me, first I prioritize. Then, I go at it. Rarely do I finish my "to-do" list so the least important things carry over to the next day. Also, I tend to be a perfectionist. That can be both good and bad, but it usually causes me to take longer at certain tasks. I rarely cut corners. Also, sometimes I delegate what I can.
I stay productive by getting proper nutrition and hydration, as well as doing exercise and meditation. We must take care of ourselves before we can take care of anything or anyone else.
True, you should always start with the most important / hardest things on the list because then they get done for sure, and also you're most productive a few hours after waking up!
Yes. I have my most energy in the morning, too.