(2n + 1)^2 is not 4n^2 + 1.
And I think he was more concerned with the remainder you get when you divide by 4, not the expression in terms of fractions.
You're getting there. What's the problem with n being 10.5. Look at your original assumptions. What are you tacitly assuming about the original n balls?
Look, you're essentially supposed to say, "Suppose Bill is right. Suppose 7/(n + 7) = 2/5. Then such and such would follow." Why would the conclusion be a problem?