[Disclaimer: this is a rough idea, not a tax code. Almost certainly needs further thought to make sure there aren't any unwanted incentives for strange behavior.]
> 1/ Shareholder and CEO. My company generates €1M in profit. I pay myself a salary of €1M, and pay a flat tax on my salary. The business pays no tax.
Right.
> 2/ CEO, but no shareholder. The company generates €1M in profit, and pays flat tax on that €1M. Dividends flow to shareholders, and they're taxed again?
Dividends don't magically appear out of nowhere; they're paid out of profits, just like a salary. (Some companies pay out the majority of their profits as dividends, others pay none at all.) They'd either get taxed at the business because they're not paid out, or taxed at the shareholders because they are, not both.
As much as I'd hate to point at it as an example of anything, think VAT, with its notion of "where was the value added", but on the income side, as "where was the profit extracted". Either a VAT-style sales tax (with less doublespeak) or a flat profit tax seems like a workable model. And much like VAT's presumption of paying sales tax unless you show that you bought something taxed, the presumption is that you're paying tax on every dollar of income/revenue, unless you provide evidence in the form of having paid that dollar to someone else who then paid tax on it (or who provided evidence that they paid it to ...).
Interestingly, that then means that if you take money out of the country and out of the economy, you get to pay all the taxes on it at that point. That seems like a nice side effect.
The structure ends up looking a lot like VAT, with a key difference: it affects all transactions, without requiring any kind of code defining taxable/non-taxable items. But apart from that, I think your questions just demonstrated that a profit tax effectively turns into VAT.
Half of the tax code for a flat tax on profits would probably end up being absurdly precise definitions of what "profit" means. (Then again, the other half, or the whole tax code for a flat tax on income, would probably end up being absurdly precise definitions for what "income" means.)
(To preempt the obvious response: yes, there are companies, such as the one-off shell companies constructed to make films, that intentionally construct their books such that they don't actually make any profit. However, that money has to go somewhere, and it'll get taxed there.)
That said, it wouldn't be the end of the world to just make it a flat (and much lower) tax on income rather than on profits. That'd be a much simpler system; it would, however, be much more of a drag force on the economy, like the current tax system. Hard tradeoff.
> 3/ What do you do with foreign companies that levy corporate tax, and then declare a dividend? Do you tax it again at the flat tax in the hands of the shareholder?
[I'm assuming you mean "foreign companies in a jurisdiction that levies corporate tax"?]
The situation definitely gets more difficult with a non-self-contained economy and multiple tax structures. It's a flat tax, so it shouldn't have a complex set of deductions and loopholes, especially that only apply to large multinational businesses. So yeah, I think you'd have to just ignore the foreign corporate tax, and treat the money paid as a dividend as income entering the economy.