1)What is the purpose of public schools and education? 2) Does free education mean unequal education? 3) Is our public education system still segregated?

My only experience in free (and low cost) education narrows down to MOOCs and Webinars. These modern offers give academically unattended communities and demographies the possibility of studying topics they are interest in applying themselves to, and that are not necessarily physically accessible in their regions. Free education is just the result of a modern democratization of knowledge.

Within the context of United States history, the purpose of public schools and education is to create a citizenry that is educated enough to vote. Remember that having a literate and informed populous is required for any successful democracy. As the ideals this country was founded upon hold that sovereignty lies with the people, the government provides education to the public. 
The second element of your question is more subjective and dependent upon how one defines "free" and "unequal." In the classroom, education is free to the student; the student pays no price of admission. However, education is funded through taxes. Some funding comes from the federal government, and some comes from state property taxes. Theoretically, free education is not necessarily unequal. However, in practice inequality occurs.
Many argue that public education today remains segregated. Some point to property taxes, emphasizing that economic segregation in communities creates segregated schools. In other words, the fact that schools in high-income neighborhoods receive more money through property taxes than low-income schools results in an economically segregated public school system. Many areas in the United States remain racially segregated. In such areas, racial segregation in schools results as well.