Why emitter follower is so noisy and distorted in large signal?
Emitter followers are perilously close to being UHF oscillators; it is quite possible for them to start oscillating during only a small part of the wanted signal cycle; this appears as periodic noise and may also cause the observed distortion. (It is usually too high frequency to observe on a scope!)

The usual cure is a small value resistor (experiment with 22 and 47R) in series with the base, (aka "base stopper") as close to it as possible.

What happens is that the device capacitances in conjunction with the inductance of the base connection cause a parasitic series-resonant circuit, and Cbe provides positive feedback, the emitter current providing the power. The base stopper lowers the Q of the resonant circuit to prevent oscillation.

(It is also possible, as Andy AKA is hinting, that R3 is too high in value and this is clipping the -ve peaks)