As we all know, the hex file is the heart of our application code which will be programmed into the microcontroller's flash memory for execution. My doubt is before the execution of this hex file, will it be verified by a microcontroller or it will just execute once all start-up processes finished?
Disclaimer: Because I don't know all microcontrollers, this is not a complete answer.*
The flashed binary executable will just be executed.
Some microcontrollers check for a certain value at a fixed address to decide whether to start the built-in bootloader or a flashed user program.
If you need the user program to be checked, you will need to implement this yourself. I have worked with such systems, it is quite common, especially in safety-related environments.
Concerning the format of hex files:
Intelhex as well as other format like SREC are human readable text representations of binary data. The common reason for the checksums in these formats is to ensure data consistency during transmission, which was done via unreliable channels back at the time when the formats were invented.
Another advantage is the limitation to 7-bit ASCII characters that can be transferred losslessly via old internet protocols.
However, the "real" contents, the binary data, is stored directly in the flash memory of the microcontrollers. Checksums might be used by the receiving software (for example the bootloader) in the microcontroller when the user program is to be flashed. But after flashing they are gone.