It depends on the meaning of the word that is followed by "skill".
So, if you want to say that "I have skill in persuading people", you should use: noun + skill.
"I have persuasion skill."
But if you want to say that "I am good at persuading people", you should say:
"I am persuasive."
And about negotiating and negotiation you should first know about the meaning of them.
Negotiating: "The act of discussing something to reach an agreement."
Negotiation: "The process of discussing something to reach an agreement."
So, the right choice is "negotiating skill".
And communicative and communication:
Communicative: " The ability to communicate in a language, especially a foreign language."
Communication: "The act of communicating and expressing ideas and feeling with people."
So, communicative skills is quite different from communication skills.
Here is an example:
"Doctors don't always have good communication skills."
And about the last one you should know problem-solution is not a noun actually it is an adjective.
Problem-solution (adjective): "A piece of writing with a problem-solution structure mentions a problem and then a solution to it."
Problem-solving (noun): "The process of finding answers or solutions to problems."
