- The company's directors / investors almost certainly know about this. It's probably a big issue, and a huge pain in the neck, for them. That kind of employee turnover must be well known to the board. Being client-fired by three different design shops probably came to their attention as well.
- If the board fires the CEO, one of them has to take over her job immediately and turn the company around. For that reason, it's very unlikely they'll fire her. They could also tell her to hire her successor, but if she won't cooperate there's not much they can do to boss her around.
- They could withhold future investment. That will probably sink the company unless you're cash-flow-positive.
- You didn't cause this problem, you can't control it, and you can't cure it. This bad behavior has nothing to do with you, except you have to witness it and catch it on the chin sometimes.
My point is this: you do not need to carry the burden of fixing this problem. That's good, because you can't fix it.
You can defend your own sanity. One strategy might be to consciously redirect any anger you feel towards her in the direction of compassion. For example, think to yourself, "wow, her personal problems must be very serious if she's taking them out on Joey."
Another strategy might be to make an "I" statement. For example, "When you shout at me in public, I feel humiliated. This makes it hard for me to concentrate on my work. Please, if you have a problem with my work speak to me privately about it." The formula: name the behavior. Name your reaction to it. Name the consequence to her. Ask for a change.
She isn't going to smack her forehead and say "what was I thinking? I'm sorry! I won't do it again." She is going to resist if you confront her with an "I" statement. That's OK. Let the resistance roll off you.
Finally, evaluate carefully what is keeping you in this company. If you have strong reasons for staying, concentrate on those. If you don't, it's resume time!