How do i establish my NY registered LLC as an office in NH so that I can
use a room in my apartment as a tax write-off? (I want the business to
remain registered in NY, just want my office space in my apartment to
be tax deductable...the rent...I know this gets tricky)

I'm not a tax expert, but I don't think you can get tax deduction when
all is said and done.

If your LLC rents the room from you personally, the company picks up a
deduction as a business expense.  Unfortunately, you personally have
to pick it up as rental income.  So you have a deduction on the
business side and earnings on the personal side.  Since the LLC net
earnings flows through your personal return on the K-1, the net
efffect on your personal income tax return is zero.

By the way, the above comment assumes you are renting this apartment.