Over the past couple months, it has become very clear that there is a culture of distrust and conflict within the company, and a lot of that feeds out of leadership relations with our CEO. The CEO is fairly egotistical - assumes they can do everyone's job better, and employs verbal abuse to make that clear. Whenever they get a scent of something going wrong in the company they will swoop in and criticize the people involved, give some off-the-cuff opinions on how they'd do it better, and then disappear to the next tirade before anything is actually fixed.
This has led to most of the company tip-toeing around the CEO and trying to just do work without pulling CEO attention. New projects and initiatives usually have a discussion topic around how to NOT get the CEO involved, or how to keep it minimal if necessary. There's back-channel talk all over Slack complaining about the CEO's most recent whatever.
The easy route here would just be to keep my head down and probably start looking for a new job... but is there a better way? Leaving this problem unaddressed probably means the company will eventually implode - what's the best way to get ahead of a problem that is sourced at the very top of the company?
I have seen this when I was a consultant for many companies and startups. The common trend is the companies would work for some period of time because of the good people around the CEO making it work. The reality is all that does is delay the CEO's failure and makes them think they are doing it right. Eventually though, people leave, new people come in and the amount of time they spend trying to fix things gets less and less to the point the company can't tread water and the cracks begin to get too big to stick a finger in.
BTW -- I've seen this with good CEO's too who were too passive. Typically there would be an exec or manager that was destroying the business and the CEO was too hands off/passive, so it isn't just a CEO that can cause this kind of damage.
My 2 cents, find a new place to work as there is little you can do to change the situation in the role you have. If you want to try and help fix it, find a high visibility project or issue to recommend/bring in a skilled & respected consultant who has direct interaction with the CEO, board or investor(s). It may still blow up, but the consultant can say things and push in ways employees can't, and if they have interaction with the board or a relationship with an investor they can backchannel pressure in ways you could never do (easily/safely) as an employee.
Look for a chance to engage those people if you can (at half the size of your firm it is usually easy) and try to feel out what they think about the CEO before you consider "blowing the whistle".
(This is a place where "digging your well" is a great idea; before you have a problem you should develop relationships that bypass the chain of command.)
It's very possible the people above the CEO are aware of the situation, it's also possible that they aren't. (Some people are nice to superiors and rude to subordinates)
Whatever you do be sure to keep your integrity and never "lose your shit" no matter what happens.