Can an immigrant without a work permit do odd jobs?

The correct answer is No.  Legally speaking, anyone who is not authorized to be in the United States should not be here.  Secondly, not all authorized foreigners in the United States -- tourists, business visitors, students, spouses of many categories of nonimmigrants -- are allowed to work, or have severe restrictions on how they can work and their visa will typically spell that out (for example, some students can work on campus for a limited number of hours each week).

In the eyes of the law, there is no such thing as odd jobs, because as long it is a job (meaning you get paid for it), you have violated the terms of the visa or broken yet another law (if you are in the country illegally).  That is why, if you babysit or clean someone's home or engage in similar other tasks, you need to be authorized to work, and based on your income, file a tax return (even if the income was earned illegally).