Could one make a specific person shoot someone dead using the deathnote?
Since you wrote both Mr.A and Mr.B's name in the notebook
and their deaths (which is caused by the book) correlate with each other, this would be possible. writing their times of death would increase the chances of this working.

for example you know both Mr.A, and Mr.B meet everyday at 7:00
you could just make it so that Mr.B gets "influenced" at 6:30 and Mr.A gets shot at 7:00, making sure you write Mr.A's fate first on the death note.