When in doubt it is always safest to assume you are dealing with a bearing wall in these situations. From your description it really isn’t possible to positively know if what you are describing is adequate. From what I can sort of tell, what you are planning doesn’t sound too kosher. I can’t see it from here though… Is there a wall directly above this wall on the first and/or second floor/attic? This usually means it is bearing. For the floor to carry the full load of having a stairwell opening you typically need to have a double joist on either side of the opening and a double header carrying the adjacent floor joists. With TGI’s this usually involves special hangers and maybe special blocking too. You should really have someone qualified look at it.