Ques 1 Preposition
Because of his distrust_________ everyone, he could never gain good friend.
The sentence would be:
"Because of his distrust of everyone, he could never gain a good friend."
Explanation:
The preposition of is used with "distrust" to indicate the object of distrust (e.g., everyone).
This makes of the correct choice.
Ques 2 Preposition
Einstein was awarded the Nobel Prize of his ____________ the theory of relativity.
Ques 3 Preposition
The way he is currying favor ________ his rich neighbor is sickening.
Ques 4 Preposition
His tendency to pick up a quarrel __________ petty things lost him a number of friends.
Ques 5 Preposition
The poor women begged __________ bread. We all begged pardon ________ her.