Did the 1960s Supreme Court strengthened the rights of accused criminals?

it gave rights to people like the right to remain silent (you don't have to answer if you don't want to) and the right to a lawyer

Yes.

Fourth Chief Justice of the Supreme Court, John Marshall, who served from 1801 - 1835.

No, not usually. New York State, which names its state trial courts "supreme courts," may try either criminal or civil cases. Texas has two final appellate courts that operate at the supreme court...

In the United States the right to counsel in federal cases was established in the Bill of Rights. That right was gradually adopted in state courts. It was extended to all states by the Supreme Court...