Traveling separately is ok. Extending (as in getting 180 days more on the same entry) separately is likely to cause issues. There has been at least one report of a dependent who could not extend their stay at the immigration office, because the main visa holder had been out of the country (without their dependent) and could not extend their permit to stay themselves (too early to apply for an extension). I cannot remember which office it was, but they definitely insisted on the main visa holder applying for an extension too, in order for the dependent to be able to apply for one themselves.
Not only my experience, some embassies/consulates even mention it in their requirements. Proof of payment is pretty clear. I agree that there is no minimum requirement for said income, but income is pretty much on the menu (meaning unpaid work would not qualify), even if they don't always require a proof of income.
And again, in the cas we are dealing with here, what could this applicant show as a proof of economic activity? Apparently there's no business registered, no contract, etc.
1. Many embassies/consulates ask for proof of income, especially for the workcation scheme, but not only. Taipei is known for asking for proof of income, even for the soft power route.
2. How will this specific applicant prove employment then? Apparently he's a trader, has no business registration, no contract to show, etc. Income is his only way to prove his economic activity.
There's a risk applying as a trader in the region, I think. If you had a business registration and applied in London for example, that would be another story.
Why don't you submit a soft power application instead?