can you prove it is closed under addition? guessing in mathematics is not the best strategy...

by your own reasoning (provided it is correct), B) is not the answer you're looking for. it's far less important to get the right answer, than it is to know HOW to get the right answer.

for the second question, a point in L is either of the form (a,a) or (a,2a). what happens when you add two points that lie on different lines?