Yes, the STM32H743 supports this, as you can read in the reference manual RM0433, section 55. Please check all the 82 pages related to that interface.
No, the MicroSD requires its own voltage, i.e. maybe operates directly at 1.8V. As an example running at 3.3V you could also check the schematics of the evaluation board.
Good luck!
/Peter
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