Asking potential advisor about their post-interview opinion on my profile for PhD application
Typically application deadlines for fall admission are around December 1st; this timing gives the admissions committee time to review applications and have students interview at the university in Jan-Feb, with admission decisions coming in Mar-Apr.

I think the graduate coordinator saying spring admission is unusual might be a polite way of saying they don't really admit for spring; if you are applying to a program that is at all competitive, it isn't about not having red flags on your application, it's about being a stronger candidate than all but a handful of applicants.

If the professor you contacted isn't making a truly special effort to get you in for the spring, I don't think you should hold out hope for that. So far, everything you said - you had a nice interview, she contacted your references - sounds typical, not like she has taken an unusual interest in you that would allow you to bypass the typical admission procedures.

(note: you said you had your interview already; I'm assuming this was an informal interview with the professor you mentioned, and not an official interview conducted after you officially applied to the program)