1) Shouldn't there be additional diagrams that look like A2, but have the derivative on one of the internal lines?

Yes! However, note that by momentum conservation the internal and the external line have the same momentum, therefore the action of  $//<![CDATA[ \begin{array}{l}\partial^2\end{array} //]]>$ on the internal line gives the same result. There could also be diagrams like A2 with the derivative acting inside the tadpole, but those, as many others, aren't included among the possible choices and don't contribute to the result required in the exercise.