will I friendzone myself?

The friendzone is a strange thing, you could still be friends with her and not be friendzoned but it's really just all in your mind. Definitely use the time to sort out things in your personal life before making proper moves (e.g. asking on a date), as you said it's best not to use her as some platform to try and make your life better. Having said that, it doesn't take much effort to show a girl that you like her. If you talk to her, just be interested in what she has to say, ask questions and make yourself an interesting person. Do compliment her from time to time, just drop hints that you do like her, it doesn't have to be serious stuff like staring into her eyes (which is actually creepy btw) but just the small things that she might start to think whether you like her or not.

From my experience, it's in action which leads to friend zone. And I'm not sure, once a girl sees you as a friend then it's very unlikely you will ever date her.

I agree, no point in asking her out if I'm not happy with my life. It will just make things worse for me.

Drop hints like tease her? Is that what you mean?

You can be friends with her but not be friendzoned, that's why I say the friendzone is a strange concept that many people seem to get confused over. You can tell her how nice she looks or compliment her hair, just the little things like that really. Yeah you can tease her a bit too Just don't make yourself too available really, don't be always there to help her with her problems. Eventually you will have to step things up though, after a few weeks maybe.