The nature of the stock market demands that they never self regulate. Companies are incentivized to meet the demands of shareholders, which means more revenue streams, regardless of the effects on consumers.

When, in the history of anything, have companies self regulated properly? It always ends up at the October 1929 stock market or worse.

Why do we need big government to prevent us from stealing or killing each other? Maybe the whole society should self regulate?

This is a straw man. A government can try to stop theft and violent crime without building regulatory moats around incumbent companies.

Though in many places governments aren't great at preventing theft and violence. That's why even in developed countries, many companies hire private security, citizens learn self-defense, and some people carry pepper spray, knives, or even firearms.

Because competitors must play by the same rules. Self-regulations are internal rules. When one wants to put a rule on that will drive up cost or decrease profits, they will only do it if the competitors do the same. That is why they need a 3rd party regulator, where they can go and agree on rules.

I think because democracy also implies a separation of powers, so that you have the players watching each other in order to prevent any player from becoming too strong.

Again, there's a Soviet counterexample of army/party/intelligence service.

Even in the case of a dictatorial and lifetime head of state, separation of powers is baked into the cake by groups with common interests, sometimes organically, sometimes on purpose.

Seperation of powers is necessary but not sufficient requirement for maintaining democracy. In classical Athens they didn't have that, so modern philosophers take pride that we do have it.

Of course any sufficiently big organization does have competing departments with conflicting agengas, but also there are nuances.

Companies will do anything to earn a profit. Asking them to self-regulate is orthogonal to the goal of maximizing profits.

Should we allow chemical companies, oil companies, and power plants to regulate themselves? What about drug companies?

The counter example is the way the Soviets dealt with chemicals, oil, power.

Personally, I think that the government vs. industry split becomes rather academic when you are speaking of monopolies, especially monopolies that spy on you. I expect that there's a slow motion merger into a GoogleFacebookMicrosoftApple.gov body with the companies acting as cabinet branches.

why should anyone trust companies given the last few thousand years of history?