Does the Czech business residence permit let me take a regular employee job too?

The long-term residence permit for the purpose of doing business is purpose-specific: it authorises you to run your trade or business, be self-employed, or act as a company officer. It is not a general work permit and does not by itself entitle you to take up ordinary salaried employment with a Czech employer.

If you want to work as an employee you would generally need an employment-oriented permit or to satisfy the conditions that allow employment under your status. Operating outside the purpose of your permit can undermine your legal stay and any future permanent-residence application. If your plans involve mixing self-employment with employment, confirm what is permitted with the Czech authorities before you start, and check the official source for the current rules.