Yes, you can remove the redundant "may"s and "affect"s.
In the first sentence, you would need to insert an "and" before the final clause. This would change the meaning of the sentence slightly. (The original sentence has an implied "and/or" between the comma-delimited consequences, whereas the revised sentence has an explicit "and" between the consequences.)
1b. His bad behavior may expose our secret, destroy our relationship, and affect me emotionally.
In the second sentence, you might want to eliminate the word "the". Keeping the word "the" restricts "the other people" to a group of people that has already been mentioned.
2b. His bad behaviors affect me and the other people.
2c. His bad behaviors affect me and other people.
In these examples, the redundant "may"s and "affect"s make the sentences clearer. I think they also make the sentences more poetic, more memorable, and more powerful.