Is it permissible for a teacher to explain the curriculum in detail to a private student whom she is teaching to memorize the Quran for a fee, when the teacher herself will be examining this very student at the institute? And is it permissible to continue teaching her to memorize for a fee?

There is no harm in continuing to teach the student for a fee, and explaining the curriculum to her in detail or summarizing it, as long as it does not involve cheating or betraying trust, such as limiting the explanation to anticipated exam questions or choosing easy sections for her in the test.