He however is not responsible for her actions.

What kind of question is this? She hexed him, because he was being a jerk. He ultimately deserved it. Slughorn was simply interested in her talents.

^...I don't remember him being a jerk so i checked ..and
"He saw me hex Zacharias Smith, you remember that idiot from Hufflepuff who was in the DA? He kept on and on asking about what happened at the Ministry and in the end he annoyed me so much I hexed him..."-Ginny
Okay now i don't think hexing (or punchin or using any type of violence) againts people who keep asking you the same thing over and over again and piss you off is right . I've never seen anyone try to punch/hurt someone for asking the same q way too much and being annoying , i've seen people snap or yell or insult , or just stand up and leave and tell the annoying person to "shove off' . So i honestly don't think what she did to him was deserved , he wasn't mocking her nor attacking her , he was being an annoying idiot ,do you punch/hex all people that piss you off? Maybe you would , Ginny would , and maybe even i would if i had magic abilities , but the right thing to do is ignore that person or just leave.

Ugh okay i am being annoying in all this trying to prove that Ginny's actions are wrong but i just want to show that she isn't a person that is that likable, but a sometimes very rude and the high school mean girl person, that still has some traits that are likeable- like bravery,loyalty to her friends and etc .

^We don't see the scene. Maybe she did initially keep asking Zacharias to shut up, did snap or try to walk off. And knowing Zacharias, he was probably quite insulting/insensitive in the way he questioned.

Obviously it was Slughorn's responsibility. Ginny didn't force him to invite her into the Slug Club instead of punishment. He was just interested in the cursing talent she'd displayed.

Why would it be Ginny's fault?

^ Because she's the one who did the hexing. No one grabbed her wrist and forced her to do it.

Exactly. It bothers me a lot that Ginny got rewarded for hexing a student rather than punished. She deserved a punishment.