Can you actually focus on how birth leads to aging, death, suffering, stress as direct experience in dependent co-arising? Can I focus on the link between birth and aging, death, suffering, in dependent co-arising in my own mind? Or do I have to know how birth leads to aging etc. intellectually.

Does birth in dependent co-arising appear in the mind?
Dependent origination is the truth that the Buddha himself saw under the Bodhi tree at the moment of his enlightenment in the third watch of the night. This after recollecting countless past lives in the first watch and observing countless beings being born and passing away in the second watch. So yeah, it probably entails a lot of abstract thought about past and future. 

This is controversial, though - many Buddhists would have you believe that DO is an entirely momentary teaching to be understood as referring to a single moment in time, rather than multiple lifetimes. This seems unwarranted given the context mentioned above.

So, if you want to become a Buddha, then full understanding of DO is essential. Unfortunately, it seems to come after (at least) four uncountable aeons plus 100,000 great aeons (great aeons being the smaller of the two types and perhaps equivalent to a big bang).

Ordinary individuals are not required to understand DO in its entirety, though it is useful as a framework for understanding reality intellectually. In terms of practice, it is enough to understand that all arisen phenomena are impermanent, suffering, and non-self. This removes any craving for arisen phenomena and frees one from ignorance, thereby severing the connection to becoming and birth.