If you worked with an obviously mentally unstable person....

Why have you not already discussed it with them? This is entirely inappropriate, and they need to be made aware. It is not only a nuisance, but it could also be a hazard.

Don't diagnose her. Avoid terms like "mentally unbalanced, unstable" when discussing your concerns. After all, the reason why it is happening is not relevant to your concerns. Your concerns are that her actions are placing other people in potential danger and subjecting them to unprofessional workplace behavior. List the issues, the behaviors you see, and set an expectation if they reoccur, her employment is in jeopardy.
I can tell you she would have been fired at my workplace for throwing things. Grownups do not throw things at each other unless they are playing baseball and it is a part of the game. That kind of action is not acceptable in the workplace. Don't give her a pass for bad behavior because you think there might be a mental health reason. Not your call, and even under accommodation regulations, you do not have to accommodate dangerous behavior.

This.

Document every weird episode.

Tell upper mgmt every time.

Let them decide to fire her.

You need to go to management with facts about her behaviour, do not mention anything that is supposition on your part. As the boss on that hall you are responsible for everyones safety, you could be called in to question if someone gets hurt.

I'd be fired for throwing things. It's classified as workplace violence. It's even in the preventing workplace violence training video. Actually first offense I'd be written up and required to do some sort of counseling, training or something. Second offense - out the door with escorts.

Yes, what everyone has said: documentation is extremely important. But document dates and inappropriate behavior, and let HR/Management make the call. Good luck - I worked for 9 years with an unstable person, though not physically threatening, and was so relieved when yet another meltdown finally made our manager take action and have her transferred.
Our workplace is sooooo much better without her, even working shorthanded since she left in Feb.

The society I live in now it's actually just the opposite.

Then you can wipe you hands free of her if she is still under probation. If she goes to her union steward and claims to have a mental illness that does not give her a pass. I am speaking as someone who has a chronic mental illness and it is her job to stay mentally healthy not your's to accommodate her. That being said she may fall under the disabilities act (whatever you call it...federally here we call it Duty to Accommodate) and that may have some clout with you, but she will be forced to disclose an action plan how she plans to deal with her issues here on in. IF SHE DOESN'T then chances are they will escort her to the door.

Yup. If she throws scissors again (or whatever) and god forbid, someone is actually harmed next time, the company is going to have a huge law suit on their hands because they knew about the risk she posed and did nothing about it. 

Very scary!!!

Yup! Get HR involved today.