Yes, they are physically distinct, and I think that each member sort of "specializes" in his own area. I see the holy spirit as like the miracle specialist, the Lord Jesus Christ as like the forgiving, gentle character, and the Father as like the one who grants justice, and is like the warrior ( Not that the Father loves war, but he is like the tough guy). But if we were to some how put the Father, Son, and Holy spirit in like the same scenario, I bet they would each do the same thing. Although physically different, I believe they share the same spirit/soul.
Your questions are hard! I haven't even read the whole bible yet, but I will try my best at these questions. You are asking if the holy trinity sort of divided into three parts right? Well,from what I have read in the bible I can't even find the word trinity. I guess the whole "Trinity" is like the ultimate team or something, and the three member are the Father, Son, and Holy Spirit.
I want to say no... It seems in the bible that each member of the trinity have the same capabilities and wisdom, but there is no "I" in team hahaha. So I guess during the bible's time, two members would just sit up in heaven while the other one did the work that each were fully capable of doing. But then again if each have the capabilities of the other, then one is no more powerful than three... I guess there are three though to reach the humans in different ways I am guessing. I hope somebody on this thread with more knowledge than me can clear this up.
But! What I know for sure is why christians believe Jesus was greater than a man, to simply answer this thread makers question! GEN 1.26 "Then God said 'And now we will make human beings; They will be like us and resemble us...'" God (the Father) is saying us, like there is more than one person there. Which means that God is not alone in the creation. God was accompanied with the holy spirit, and Jesus! Yay! Therefor if Jesus was there during the creation, then he must not be a mere human if humans were made like him.