Patur,
1. I wrote that the person who told me is a rav who learned in YU. However, you are correct that he was not a personal talmid of Rav Gorelik. In any case, considering the fact that the same types who want to besmirch Rav Kook are also denying the extent of Rav Gorelik’s connection with YU I do not find them reliable.
2. Apparently Rav Kook was still strong enough to speak with Rav Soloveichik in learning as he instructed Rav Tzvi Yehuda to attend all of Rav Soloveichik’s shiurim (the reason he was here was that he was a candidate of Rav of Tel Aviv – in the end the electors decided that he was too young and chose Rav Moshe Avigdor Amiel).