Only in case the former Hokage somehow isn't able to choose his successor himself!
Hiruzen was chosen as the leader by Tobirama out in some forest while on a mission during the war. Hiruzen chose Minato.
The Hokage has the full power to decide who takes his place!
Not true. Hashirama wanted madara to be hokage but the the citizens didn't agree. Even when Hishirama said Madara could become the 2nd Hokage Madara knew that the citizens still wouldn't agree to it. That's part of the reason why he left. Becoming hokage is a group decision. Itachi said it himself. One becomes hokage only when the people accept him. Hirusen was lucky he had The 1st and the 2nd as teachers, that most likely played the biggest role in him becoming kage. Same goes for all the other kage.
1st and 2nd related
3rd being a student of 1st and 2nd
4th being a student of the student of the 3rd
5th being a student of the 3rd and granddaughter of 1st, grandniece of 2nd
6th(naruto) student of the student of the 3rd, son of the 4th
7th(konohamaru) student of the 6th, grandson of the 3rd
Candidates:
Kakashi: student of the 4th
Orochimaru: student of the 3rd
Danzo: student of the 2nd(possibly 1st as well)
Face it itachi doesn't make the line up.