There is categorically no minimum income requirement for self-employment.

Example: if I buy a shop and run it and work 50 hour weeks but in the first two years of trading I don't make a profit and pay myself very little, I'm still clearly self-employed and genuinely working, even though I'm not paying tax because I'm not paying myself a salary.

Self-employed or employee when owning shares in your own or your partner’s company

If you own the majority of the shares, or the company is co-owned with your partner or another family member, then the Home Office will expect you to provide evidence as self-employed.

You can include your employment evidence, but it is not a typical employer/employee relationship when you are the company owner. There is no contract of employment (written or unwritten), no employers liability insurance, no genuine employer/employee relationship.

So, you can declare yourself as an employee if you wish, but you should also be declaring self-employment and providing evidence of this.

Companies House website lists all company owners and directors, so anyone can check the ownership of a company in seconds.

See Evidence of self-employment to know what to send.

Self-employed and overseas income

I was billing clients through my father’s company & paying taxes in Finland, then I registered as self-employed in the UK.
If you get an income from overseas in this way, it can only be counted towards working in the UK under the EEA Regulations if you have registered with HMRC.

You do not have to pay tax in the UK if you are paying tax elsewhere as you were, but you need to register and complete a self assessment tax return declaring the tax paid overseas.

Rental income from property, investments, etc.

Income from a rented property is unlike to be considered as 'self-employment' as it is really 'income from an investment', which is self-sufficiency. This type of income is not classified as earned income.