She should never hit, kick, push, etc. her date (or anyone else). Nor should she be breaking things on purpose.

Some people will "play fight," such as if she grew up wrestling her brother or trying to push him onto a couch. However, this play isn't meant to cause harm, and both parties consent to it.
Some people kick someone under the table, or play slap them, if they say something rude. However, these shouldn't cause real pain and shouldn't be upsetting to the recipient.
If she breaks things when upset, notice (1) whose things she breaks, and (2) who cleans up. If she's truly losing control, her things are just likely to be broken as yours. And if she's remorseful, she'll clean it up herself. But if she only breaks your things, and then has you clean it up, it's a power play.
Abusers are often very good at hiding their true selves, so they might not give any obvious signs until you're in it deep. Even then, they'll keep you questioning it. This is the nature of the abuser, and it's not your fault for not seeing it.;
, It's highly unlikely that everyone this woman associates with is irrational—the problem is most likely her.

Calling other people "crazy" is a red flag in general: it's dismissive and shows that the person may not think highly of people who don't act like the person wants.
 If she often switches jobs, friend groups, or living situations, then there's likely an underlying problem. Sometimes the problem isn't with her personality (for example, her awful parents kicked her out of the house at a young age and she had trouble finding a place to live for a while). Other times, the problem is her.

A long history of making enemies often means trouble.
 Some people have decent pasts, and some people have hard ones. If she's gone through a lot of hardship, she may be hesitant to open up to you, because it's painful and difficult. But if she's acting secretive or shifty, it may be that she has something to hide.

There's a difference between hiding genuine pain because she's not ready to share, and hiding secrets because she knows you wouldn't like what she's done. Are her motives honest, or manipulative?

, Some women are more confident than others, and it's normal if she jokingly compliments herself (such as if you say "You're pretty" and she says "I know" and gives you a flirty smile). But constantly building herself up could mean that she has self-centered tendencies, or serious self esteem issues that hamper her ability to have a real relationship.

Notice the difference between joking bragging and excessive bragging.
 It's okay for a girl to be forward if she likes you. However, if she tries to integrate herself too quickly into your life, without letting you have a say in it, then she may be obsessive.

Do you feel overwhelmed by how much she tries to contact you? Does she try to get involved in every aspect of your life: friends, family, etc.? A secure woman will feel comfortable letting you have some time to yourself, because she trusts you'll always come back to her.
If you feel pressured, tell her that you want to take it slowly. Say you'd be more comfortable that way. If she listens, then she's excited but clearly respectful of you. If she refuses to accept your boundaries, then it may be time to break it off.
 A good girlfriend is honest, even if she think you might not like what she has to say. She shouldn't be hiding things, constantly dodging questions, or keeping big secrets.

Sometimes people will tell white lies (like "yes, I like your tie"). This is ordinary and not a big deal.
As you two become close, she should be open about exes (if they come up), and whether she's seeing them at all in her life.
Don't expect her to share everything with you. She doesn't need to tell you what she ate for breakfast, or the precise details of her time with friends. A good relationship has a healthy balance of truth-telling: saying the things that are important, and leaving out minor details that don't matter.
 A mature person will admit she's not perfect, and that past breakups may have been partially her fault. She usually won't talk extremely negatively about her exes, unless they were truly awful or if she's still caught up in her feelings.

If she badmouths her past dates, she may talk the same way about you later.
Consider if she conflates facts and feelings, or calls names. "I felt really pressured and uncomfortable, so I broke it off" is different from "My ex girlfriend was an obsessive stalker-like creep."
A mature woman might say things like "We were a bad match" or "He wanted so much space I didn't feel like his girlfriend anymore," acknowledging two perspectives. She may have had a few experiences that were completely the other person's fault (e.g. an abusive ex). But, depending on the circumstances, she won't always place 100% of the blame on the other person.
 Not all differences mean that your girlfriend is undateable. Some people have issues outside their control. As long as she never makes you feel unsafe, then she'll be a good girlfriend as long as you enjoy spending time with her.

A girlfriend with an anxiety disorder might get very worried sometimes, need reassurance, and hyperventilate over things that feel small or overblown to you (such as thinking you'll leave her even though there's no way you would).
A girlfriend with bipolar disorder may have mood swings she can't control.
An autistic girlfriend might be sensitive, act eccentric at times, and have awful crying episodes when she gets overwhelmed. She just needs to "cry it out" and rest, and then she'll be back to normal.
Be very cautious if she has violent tendencies.
Listen to see if she refers to other people as crazy: "my insane boss," "my psycho ex," "my crazy mother".
Note patterns of unpredictability.
Consider how she talks about her past.
Notice how much she builds herself up.
Consider how quickly she tries to get close to you.
Beware lying.
Notice how she talks about her past relationships.
Recognize that some disabilities or mental illnesses can cause unusual behavior.
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