Can you 伤害 a person emotionally? Can I say “伤害人” in terms of hurting someone emotionally? Or is there another, better, way of saying this? 

For example, I'd like to say 

"最后还为了伤害他把这个秘密告诉给其他的家人" 

Is that a proper usage?
Not a language scholar so may be I am wrong on this, but the "伤害" is probably more often used in terms of hurting someone emotional than physically. 

Example：in 那英's famous song "一笑而过“ there is one line reads "你伤害了我，还一笑而过". As the song is probably not about domestic violence, yes, it is perfectly proper to use "伤害" in this way you described. 

It is actually uncommon to see people use 伤害 as equivalent to physically "hurt" someone. It is too formal, a little awkward, and sounds pretty harsh.

伤害 can be used as a noun as the above answer has mentioned the phrase "生理伤害". In this usage it sounds very formal. So most likely it is used in a legal or medical document. 

The word 伤 alone can be used as a consequence of some action. For example, 他把我打伤了 (He hit me and I am hurt/injured as a result).

The sentence you gave is good and can definitely be understand. But it may help to change the wording a little bit. For example, although can be inferred, the sentence do not have a subject. You can say 最后为了伤害他，他/她/name 还把这个秘密告诉了其他家人。

其他家人 sounds a little strange and unspecific. If it is one or two people, you can list them instead (to make the story more convincing and interesting): 他还把这个秘密告诉了 爸/妈/爸妈/奶奶. If it is a lot of people in the family, you can say 他还把这个秘密告诉了 全家所有人. Technically, 告诉了全家 is good enough and it may not be literally every member of the family. But usually you would add 所有人 to emphasize (and the listener/reader would know that 所有人 is most likely not literal here). There is no need to use 其他 as it is implied. (As you can see that us Chinese are not very rigorous when it comes to language. You are generally considered a better writer if you could use less words/characters to convey the same meaning.)

Source: native speaker.