While you are correct about the EXTI0 - EXTI15 pins being configurable for a wake up, unfortunately, this particular series of microcontroller (STM32F4) cannot have the USART clock active when stop mode is on. This means that the peripheral cannot see any data. You can; however, use an external watchdog, RTC, etc... this will allow for that with your current microcontroller. There are workarounds for this.
You could use sleep mode, which just the Cortex M4 Clock and the CPU would be stopped while all the peripherals are left running. However, with all the peripheral clocks enabled you will draw more current.
If you are interested in USART clock functionality in stop mode, check out the STM32L0, or STM32L4. Both of these have that feature and it works phenomenally well and I would highly recommend these two series for a low-power application as this is what they are designed for.