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Is anything known about the level of expertise he has attained and whether he was running a small business, or working as an employee in a shop or a day laborer?
What little we know for sure is that Joseph was a carpenter:
As the eldest son Jesus under Jewish law would have inherited one half of Joseph's wealth. That would mean that he was in line to inherit Joseph's business. As the apparent heir Joseph would have taught Jesus all the skills in carpentry that he had most likely learned from his father as well as through his own experiences.
In that time the business would have been a family business and all four of Jesus' brothers would have been likewise trained by Joseph.
Those things are discernible from Scriptures, and some other writings from the period, such Josephus among others.