A complete sentence has a complete thought, it contains a subject (it mentions who or what the sentence is talking about) and a predicate (the action the subject does or is). On the other hand, a sentence fragment is incomplete because it only has a subject or a predicate. For the options you have we could say:
1. Kids have played at O'Leary Park for year. This is a complete sentence. The subject (Kids), does the action (have played at O'Leary Park for year).
2. Even my grandpa when he was young! This is sentence fragment. We know his/her grandpa is the Subject but the thought is incomplete, we don't know what he did when he was young.
3. The city wants to pave over the field. This is a complete sentence. The subject (The city), does the action (wants to pave over the field).
4. Will make posters to protest and write to our mayor. This is sentence fragment. It is incomplete because we don't know who is going to make poster to protest.
