Electronic – STM32 SDIO works only while probing clock signal

Looking at the datasheets you provided, if you want to use RMII mode there seems to be no other alternative than what you already suggested. Namely, you must connect 50 MHz clock output both to PHY (REFCLK) and MAC (RMII_REF_CLK) in order to synchronize them.

Similarly if I want to UART at 115200 Baud, does this mean my min. AHB and APB clock speed is 115.2kHz?

No. The USART performs 16x oversampling; the baud rate is given by:

$$F_{CK} \over 16 \times \mathrm{USARTDIV} $$

As the minimum value of USARTDIV is 1, the peripheral would need be clocked at a minimum of 1.8432 MHz (16 × 115.2 kHz) to run at 115.2 kBaud.

Additionally, keep in mind that using a USART divisor of 1 leaves you no room to tune it -- if your system cannot generate a peripheral clock of exactly 1.8432 MHz, you may need to increase the peripheral clock to make it possible to derive an appropriate USART clock.

Finally, consider that the USART also supports DMA -- it's possible to output data over the USART while the CPU is sleeping. You'll only need to wake the CPU for as long as it takes to convert data from the ADC buffer to the format you want to output on the USART and set up DMA.