Can VOUT of the MAX77542 provide a stable voltage without VIO being powered?

No, VOUT cannot provide a stable voltage unless VIO is powered. VIO must be above the Under Voltage Lock Out (UVLO) level for the controller to function and enable the outputs.

Can VOUT1 of the MAX77542 be used to supply the VIO pin, thereby eliminating the need for an external VIO supply?

VOUT1 cannot deliver a stable output unless the VIO (logic supply) is powered above its undervoltage lockout (UVLO) threshold. If VIO is below this level, the controller disables all outputs to prevent malfunction. For normal startup and operation, VIO must be supplied either from the input voltage or an external source, but care must be taken to ensure it does not exceed the absolute maximum voltage rating for VIO.