I'm getting really concerned about her. She's 9 years old, and already has severe anxiety. My mom and I think she has anxiety neurosis, due to her OCD-like behavior (but she's not OCD, she doesn't do things like clean just to clean, she does it because she thinks our floors are poisoned or something like that) and her anxiety.

There are many different specific forms of neurosis: pyromania, obsessive-compulsive disorder, anxiety neurosis, hysteria (in which anxiety may be discharged through a physical symptom), and a nearly endless variety of phobias. According to Dr. George Boeree, effects of neurosis can involve:
...anxiety, sadness or depression, anger, irritability, mental confusion, low sense of self-worth, etc., behavioral symptoms such as phobic avoidance, vigilance, impulsive and compulsive acts, lethargy, etc., cognitive problems such as unpleasant or disturbing thoughts, repetition of thoughts and obsession, habitual fantasizing, negativity and cynicism, etc. Interpersonally, neurosis involves dependency, aggressiveness, perfectionism, schizoid isolation, socio-culturally inappropriate behaviors, etc.

She's also starting to show signs of depression. My mom is unsure about taking her to our family doctor, and she doesn't want Elizabeth (my sister) on medication. I've suggested taking her into my next therapy session, and having my therapist talk to her a little bit, so we can get my therapist's opinion (if we should take Elle to our doctor, or if we should start her on therapy, but my mom isn't sure about that idea either. I'm concerned about my mom's uncertainty - she's done it with my anxiety/depression, and it resulted in it being much worse.

I just want to know what I can do to help, or things I can suggest to my mom. I'm so worried about Elizabeth, and it's so worrying that a 9 year old has all of these issues.

Okay, I'll make sure I tell my mom that she needs to be diagnosed. I'll tell her I'm concerned and that I think the best step is to get her diagnosed right away.