> However, I am still worried that in a few months they will realize that I am incompetent or unproductive and I will be fired.

Whoa, where is this coming from?! Jeez, is this your first job?

> Do you have any things you might add?

The note about sleep is good, you probably don't need the others. Let any of the above be driven only by your passions and interests. At work, communicate well and as clearly as possible. Show up, be positive, get your work done.