The Chofetz Chaim was vehemently anti-Zionist, as were all the great Gedolei Hador of that caliber. Rav Shach writes (Michtavim Umaamarim vol. 1: Eretz Yisroel) that when he comes before the Bais Din Shel Maalah and they ask him why he wasnt a Zionist, he will “point to the Chofetz Chaim and say “because he wasnt a Zionist”. The Chofetz Chaim’s closest Talmid, Rav Elchonon Wasserman zt’l was THE most outspoken anti-zionist of his time (“Zionism is avodah zorah, and religious zionism is avodah zorah mixed with religion” is a quote from him). And he quotes his Rebbe the Chofetz Chaim often in his anti-zionist teachiings, in “Ikvese DMeshichah”. The Satmar Rebbe quotes Rav Elchonon saying in the name of the Chofetz Chaim that the Zionists are real Amaleikim. Rav Berel Soloveichik ZT”L, son of the Brisker Rav ZT”L, used to relate to his students the Chofetz Chaim’s response when he heard of Rav Kook’s position on the chloni soccer players. “Kook shmook!”, the Chofetz Chaim said, dismissing both the man and the position. The story about the Chofetz Chaim – the paragon of Shemiras Halashon himself – is easily confirmed. The person who it happened with was named Rabbi Avrohom Moshe Gorelick, father of Rav Yeruchem Gorelick ZTL, who was a talmid of the Chofetz Chaim and a Rosh Yeshiva at Yeshiva University. Rav Yeruchem Gorelick (who was personally present and heard it directly from the Chofetz Chaim’s own mouth) said the story over publicly numerous times (including at the Yeshiva of South Fallsburg, where his son was Rosh Yeshiva), as did Rav Berel Soloveichik ZTL, Rosh Yeshiva of Brisk. (Rav Berel also related that Rav Kook said about the opening of the Hebrew University, that it is a fulfillment of kimitzion etc. – immediately the gedolim in Poland and Russia organized a protest against this chilul Hashem – and the Chofetz Chaim came in and said Kook shmook and then he left. See “Mishkenos Harayim” 3:1-108.)
And the Chofetz Chaim’s statement is mild compared to what other Gedolim have said about him. Particularly relevant in this context is a letter by Rav Elchonon Wasserman to Rav Yosef Zvi Dushinsky, printed in Kovetz Maamarim, of which a facsimile of his Ksav Yad is readily accessible, where he refers to R. Kook as a “rasha gamur.” This was the eternally loyal Talmid of the Chofetz Chaim, who spent his life disseminating his Rebbi’s Torahs. (When the Chofetz Chaim was considering moving to EY, Rav Chaim Ozer asked him who will take care of Klall Yisroel in Chutz Laaretz if he leaves. The Chofetz Chaim answered, “What do you mean? You have Reb Elchonon!”)
Regarding Rav Kook specifically, I have heard that the Chazon Ish ZT”L used to censor his Seforim by taping or marking over the anti-Torah writings in them. Of course, the Chazon Ish was more able to know what is undesirable and what is not, than the average student. If someone was the biggest Apikores and enemy of Hashem, as long as he would “work the land” of Israel, Rav Kook considered him holy. The soccer players, mechalelei shabbos b’farhesia, were to Rav Kook, “holy”. He did not mean “Tzelem Elokim” holy, but rather, because they assisted the Zionist cause they were “holy”, regardless of their status according to the Torah. Rav Yosef Chaim Zonenfeld ZTL said that he was like a person who is drunk – saying irrational, nonsensical things – and in his case, he is drunk on Ahavas Yisroel. Rav Yosef Chaim Zonnenfeld’s description of him as a “Purim Rav the whole year”. Rav Kook was considered a lone, sad case of greatness gone irrational. A more hard-line position is found in the Teshuvos Divrei Yoel by the Satmar Rav ZT”L (CM 131), where he rules outright that it is forbidden according to Halachah to follow any Halachic rulings of Rav Kook, who he categorizes as an apikores, in any area of Torah. His reasoning is based mainly on the following sources: Birkei Yosef 243:3; Responsa Bais Shlomo YD II:101; Chasam Sofer CM 163; The Gemora (Shabbos 116a).
IY”H I’ll address additional points in a followup comment.