Driving a PWM output of 10 MHz with 50% duty cycle

Can Anyone explain if I would be able to get a PWM output of 10MHZ at 50% duty cycle on Arduino uno, Is it capable of doing it?

No, I don't think so. An Uno has a 16MHz clock. Logically it would need to be at least 20MHz to produce a 10MHz signal with 50% duty cycle. But I doubt that's possible even with a 20MHz clock. Also it would be a waste of a microcontroller just to produce that signal. Look for a clock generator chip.

What connection does you question have to either LEDs or multiplexing?

If you used one of the 20MHz packages, like MightyCore for 1284P, clocked it 20 MHz, divided the system clock by 2 internally and then output it on the Clock Out pin, 10 MHz would not be a problem.