Thank you for that. I can't get the stamps any further than what I've said above. They may not be shrine stamps (well, none of them say they are from a shrine, and usually the words "shrine" are included somewhere in the stamp). So they may be personal stamps of people.
The flag, as you might know, has the name of a calligraphy group on it ("Koimizu-kai"), and was presented to Mr. Yasuhara Nobuo, who would go on to become a calligraphy artist and author. It is dated April 10th, 1943. At the bottom are names, presumably of his fellow calligraphy enthusiasts. Written on the right side of the flag is a poem, but it would take some time for me to decipher it. Across the top is 盡忠報国 (Jinchū Hōkoku), which means basically to give your life for your country. To the immediate right of the red disc (and next to the poem) is the phrase "pray for good luck in battle".
No idea if it is authentic or not. Looks like its in great condition. Yasuhara Nobuo didn't die that long ago. Could be that his descendants are selling off some of his articles?