About when and where Omosu Seminary was built.
My knowledge in this field is limited. So don't have a big expectaion. I
have checked the HP of Omosu (Kitayama) Honmonji. People in the temple
themselves appear to think that their temple was built in 1298 not in
1291 and they also think the place of Omosu Seminary was in their temple.

They say about the history of the temple,in the HP, "Nikko Shonin left
Minobu in December in the first year of Sho-ou(1288), the sixth
anniversary of Nichiren Daishonin's death. He moved to Ueno at the
invitation by Nanjo Tokimitsu and made a hermitage at Oishi-ga-hara in
October in the third year of Sho-ou(1291--this is wrong. correctly 1290,
K.K). But he loved the scenery of Omosu Maruyama and was donated a piece
of land by two stewards, Ishikawa Yoshitada of Omosu and Najo tokimitsu
of Ueno. Then he began the construction with the help of Hokke believers
in Ueno and Koizumi(a place name --K.K) in the first year of Ei-nin
(1293). In the sixth year of Ei-nin(1298), Hondo(the main hall),
Mieido(the Image hall) and Suijakudo(a kind of hall --K.K) were built.
They hung out the tablet written "Hokke Honmoji Kongen"('Kongen' means
'root'--K.K) meaning that the temple was the root to achieve the
building of the Honmon Kaidan(ordination platform--K.K) which was the
oath of Nichiren Daishonin. Nikko Shonin who spent thirty-six years in
this temple, raised Omosu Dansho(Seminary--K.K) and tried to educate
young men and to spread daimoku until his death on the seventh day of
the second month in the third year of Gen-ko(1333), at the age of
eighty-eight." from HP of Omosu Honmonji, originally in Japanese.
(http://park16.wakwak.com/~honmoji/)

This explanation is consistent with the description of SGI's Dictionary
of Buddhist Philosopy (the 3rd edition) about the year of the
construction. But it has not been verified yet what buildings were
constructed and what the Omosu temple looked like at that time .

About the time when Nichijun was appointed to the chief instructor of
Omosu Seminary.
I am not much interested in the details of the history of NST. But there
seems not to be a big problem about the time when Nichijun became the
chief instructor. SGI's Dictionary of Buddhist Philosophy says that
Nitcho (Jakusen-bo) died in 1310 and Nichijun was appointed the chief
instructor in 1317 when he was twenty-four years. So I think that there
was seven or eight-years absence of the chief instructor. I also hear
that the position of the chief instructor is not a standing one in NST.

About Iyo-bo Nitcho's participation to Fuji school, SGI's Dictionary
says "He made Nichiyo to succeed Mama Guhoji temple and went to Omosu to
surrender to Nikko in 1302. He built Shorinji temple in Omosu in March
in 1303. He wrote a letter for Nikko to Ryosho-bo Nichijo to encourage
the disciples under the persecution in Kamakura. He passed away in Omosu
around 1317."
If he had served as the chief instructor, some record about it must have
left down the ages. Even the year of his death isn't clear. So the
possibility that he had been at the position of the chief instructor
seems to be little.

Kazuo.