hiya--
for the first one, the verb "protest" doesn't need a preposition after it. you protest something, instead of protesting over something.
for the second question, the subject is "competition," not "my sister and me." the main verb in the sentence is "began," and the competition is the thing beginning. "my sister" and "me" are both the objects of the preposition "between," which is why you need the object case.
for the third question, the proper idiom is "between X and Y," not "between X or Y."
for the fourth question, "would have thrown" describes a situation contrary to fact. if this were the usage intended by the sentence, we'd need the rest of the condition to be included in the sentence (" . . . would have thrown coal overboard in poor neighborhoods if they had been allowed to.") since there's no "if" phrase, we know that the sentence is trying to describe something that actually happened. in that situation, "would throw" means that the action used to happen constantly, and would be correct in this context.
hope that helps--i'm on my way to the gym so it's a little rushed
