1. Transformer is not an active device. It cannot convert 50 Hz to 100 Hz. Your main supply is at 50 Hz. The worst a transformer can do is produce noise at 50 Hz. Your problem is not a transformer issue. Transformer just decrease (step down) or increases the voltage. It cannot increase or decrease the frequency. Also transformer output undergoes ac-dc conversion and then filtering using capacitors. The capacitors remove the ac ripple noise. Let us say your capacitors have gone bad. In that case you will get 50 Hz noise and not 100 Hz.
2. 100 Hz is not normal sound. Your main supply is at 50 Hz. If there is any problem with DC rectification, shielding, etc then the 50 Hz mains supply will manifest as 50 Hz noise and not 100 Hz. Remember that 100 Hz is exactly 2x of 50. This looks like some motor, refrigerator running in the same building producing the noise.
3. Without any input connected you should be getting almost 0 volts on the output. Oscilliscope measures the waveform. For DC it will be a straight line. For 50 Hz it will show a sine wave of 50 Hz. If it shows 100 Hz output without any input it means that some component in the AVR is oscillating and generating a 100 Hz tone. If your AVR is fine, It will show nothing. Worst case it will show a sine wave of 50 Hz indicating mains leakage. 50 Hz noise is like a hum. 100 Hz noise is like more than a hum, something less than a motorboat sound.
4. If your AVR has no issue then there will be absolute no sound with any input connected, provided the noise is not from the mains and noise is not being injected from external source