Am I right to not go out with her due to being ignored in public?

Alright, so I met this girl in practice a month ago. We liked eah other very much. She came to see me on Sunday's on my volunteer events. I asked her out last Sunday, and she rejected my date request and felt bad about it. She told me that she was busy all day all week. She wished me to ask her out as I planned to, and yet it failed.

Then she told me on AIM that she plans on ignoring me during practice just to see how I can survive without talking to her. She said she planned on ignoring me during practice within a week, and I got pissed off about it. Then we got into a fight and apologized to one another afterwards.

Yesterday night, we were blah to each other because I couldn't stand her unconfident replies on AIM. I put full effort to respond her questions and most of the time she didn't comment on my things that I talked about. We were also playing around during that time. Then I told her that she should be free around next week, and she said no and depends where I would take her out to. I told her that I wanted to take her out on the beach, a fancy restaurant, movies or ice skating on phone, then she said no. She still liked me..

Then I asked her questions like what's the point of bein in a relationship if we don't ever go out and if we ignore each other during practice. Then she said I don't know and I encouraged her to make up her mind. Then I asked her would she rather be my girlfriend or be single, an still said I don't know and got pissed off at me. She told me that she's not used to going out with guys and it felt weird for her. I apologize to her and she told me to leave her alone.

An hour after the fight, she told me that she would like to go out with me no matter if her family members would catch her in school. And I told her that I had to think about it. Her other friends encouraged me not to go out with her anymore because she can't handle relationships and make things really complicated. I just got less interested in her. Then today, I asked her when she would stop ignoring me during practice and she said she did not know. She also Sid nor know whether she would ignore me in school or not, then I got jealous. I told her that I am not ready to be in a relationship and her habit of ignoring me in practice because I can't deal with dating if were going to ignore each other in public. Then I got her heart broken and she told me to leave her alone. I have some questions:

1.) Am I right to not go out with her on a date? Or am I wrong? Why? In fact, I don't like people who ignores me on purpose.

2.) is she the wrong type of person for me to go out with?

3.) how will we settle this after her heartbroken time? And I still want to be BFF with her.

4.) I'm cheap, and kind of poor. I'm also a mamas boy sometimes. I'm behind in school, am I a bad candidate to be in a relationship? (all questions answered for best answer)
Am I right to not go out with her due to being ignored in public?
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