1. The legal basis for the government to prohibit racial segregation is on the landmark case of the Brown v. Board of Education (1954) in which it was determined that “separate but equal” facilities in schools were, indeed, unequal and a form of discrimination that instilled a sense of inferiority to African Americans children, and thus it violated the 14th Amendment (an amendment that guaranteed all citizens equal protection of the laws).
Furthermore, this issue is handled at the federal level because section 5 of the same amendment holds that the government has the power to enforce, by appropriate legislation, the provisions of the amendment.
2. Supporting religious schools with government fund is a way of promoting the establishment of a religion in the nation, which is prohibited under the 1st Amendment.
3. The Constitution does not say anything about giving the government the power to establish or control the educational system, and since the 10th Amendment states that the powers not delegated to the government and that are not prohibited to the States, are reserved to the States, then the States are free to decide the curriculum of their educational system and the structure of their schools. Now, one of the reasons that States may want to set school standard is because it deems education as elemental to form good, thinking and critical citizens that work toward the benefit of society.
4. The local government gets the power to tax people from their own constitution and this is handled at a local level because the taxes benefit the schools of their own states. Furthermore, taxing is one of the concurrent powers, meaning that both the states and the federal government can exercise simultaneously, but on different levels.