Say the country is like a big corporation. Every citizen owns one share. The number of shares always equals the number of citizens and can't be bought or sold. Every shareholder/citizen receives identical regular (annual, quarterly, whatever) dividends. This is, of course, in addition to whatever everyone does in the private market, not in replacement of it. What do you think?
Government isn't designed to generate profits, so there would be no dividends to distribute. That said, I'm not even sure I understand the problem that you are trying to solve with this proposal.
State/collective ownership of means of production would be required....such would not be called "Democratic Capitalism"....it's already called "Socialism" or "Marxism"
Yes. There's a system like this in Alaska. It works very well, if you ask the Alaskans. Every family gets, say, a thousand dollars every year from the Alaska State Permanent Fund (it's usually more than that but this is just a hypothetical example). A family of six gets six thousand dollars. The people who are building a business still benefit. That business will have a better chance of succeeding since everybody has more money to spend.