In most States, you'll need either a B.Arch or M.Arch if you intend to pursue licensure as an architect. A license is not required to work in the field in any State as long as you're working under a licensed architect (or on single-family residential or other small "exempt" projects), but is necessary if you ever want to go out on your own and good for career advancement in most firms even if you never stamp drawings. What school you get the degree from really doesn't matter that much as long as it's accredited, unless it's either a very high prestige or very low reputation school; and what state you get the degree in doesn't matter at all. I'll let others chime in on whether ASU falls in the "very low reputation" category as you seem to think it does - I have personally never heard anything about it, good or bad.

An unaccredited architecture-related bachelors will make an M.Arch program shorter than if your BA/BS were in an unrelated major and might make getting into an M.Arch program easier. Again, it shouldn't matter what school it's from unless it's a really good or really bad one.

As far as job prospects if you don't pursue the M.Arch, an unaccredited bachelors probably isn't worth any more than an associates degree - it'll basically get you a job as a drafter.