The writing from the Vilna Gaon are most likely referring to the sealed letter which the Gaon wanted opened 100 years after his passing. It can be found in the Rav Menachem Kasher edition of Sefer Kol HaTor entitled "HaTekufah HaGadolah". It was published in 1972 and is still available in print. This schedule is also discussed in Sefer Kol HaTor.
The Arutz Sheva article in question was actually quoting Rav Yitzchok Kaduri, zt"l, who said what you have mentioned. Rav Kaduri said the 3rd and final stage of the war of Gog u'Magog began on October 7th, Hoshana Rabba of the year 2001 with commencement of the bombing of the Taliban in Afghanistan.
The link is:
The information is accurate and correct. This would also follow the tradition from the Tur, Orach Chaim 490 said in the name of Hai Gaon which says:
והכי אמר רב האי שמעתי מפי חכמים כי תחיית המתים עתידה להיות בניסן ונצחת גוג ומגוג בתשרי
This view was not only held by Rav Kaduri. For those who remember, the nascent Sanhedrin produced a scroll that was to be delivered to then President George Bush warning him that he was the one foretold in the Prophets in the Tanach as being Gog mi'Gog. This scroll warned clearly that Bush needed to support Israel or G-d would deal with him severely. The following links relate to this subject:
Yes, that is Rav Adin Even Yisroel Steinsaltz acting as Nasi of the Nascent Sanhedrin reading and signing the Megillat Bush/The Gog u'Magog Scroll.
I have been looking through the Vilna Gaon's commentary on the Mechilta and have found nothing that explicitly states what is being asked. The commentary is called "Biyurim v'Haga'ot HaGra". I also checked his commentary "Kol Eliyahu" on Tanach. It is possible there is some larger, more comprehensive commentary.
What I have found is only an explanation of what the Mechilta itself says. Namely that the final redemption will parallel the redemption from Egypt. In this particular case, the splitting of the Red Sea and all the details of this are associated with the final stage of the conflict in the War of Gog miGog. This is explicitly associated with chapter 60 of Isaiah at the beginning. It also relates this to what happened with the judgement of Sodom and Gemorrah.
A little later in the Mechilta it mentions the view of Rabbi Eliezer who says that although the exodus from Egypt was in the month of Nisan, for various reasons, the final redemption will be in Tishrei. The Gra there doesn't say how he holds. He just explains Rabbi Eliezer's reasoning.
The judgement of the nations is sealed and execution begins from Hoshannah Rabbah. That is the explicit teaching of Rabbi Shimon bar Yochai as found in the Zohar on parshat Tzav.
So in context, if this judgement with Gog u'Magog parallels the timing of the splitting of the Red Sea, it would seem to be shortly before daybreak.
Looking simply at the timing, the beginning of the bombing in Afghanistan in the 2nd Gulf War started the night of Hoshannah Rabbah. That complies with the plain meaning from the Mechilta as understood by the Vilna Gaon.
This is also the explanation of how the final redemption unfolds as discussed in Sefer Avkat Rochel of Rabbi Makir ben Abba Mari, volume 1, Wars of King Moshiach, 8th sign. This is also brought in Targum Yonatan ben Uziel on Devarim 34:3.