Dino, I am going to go out on a limb and interpret that you are looking for more to do. Correct me where I am wrong.
It sounds like you are looking for a clear direction. You ask what people should be doing with their first 2 hours of the day. You also ask if others spend their time with day-to-day vs "project -oriented" work. I am speculating that this is a situation in which you are questioning your worth as far as how others perceive you.
Is there a lot of catch-up every day where you work? Does everyone have to hit the ground running as soon as they get in? Does that make you have to run? Do you know of any places in the process that can make life easier? Does it feel interesting to know that you can change certain things?
You've been in your spot for 2 years. What changes have you seen in how you work and how people see you in that time?
I guess you raise more questions than answers with your post.
I will try to answer, though:
I spend my first two hours where I am needed most, and I understand where that is by my calendar and emails or calls that I receive from internal people in my company or by what I think I should be doing (sometimes that is at the top). But if I have something else on the go and someone needs me, I might be able to give someone in my team an opportunity to deliver on it, and that might be a special occasion for that person, who was looking for something to do that they can call their own.
But that really doesn't have to be only the first two hours.
What else can you say about what you do? There is peer feedback on the way, if you can speak to some of the things I said, I would think.