It might be no good for LEDs at allWill this burn the light bulbs and melt too or can I just replace the lightbulbs?
12V Effective - who knows what the frequency and peak-to-peak voltage is. Your LEDs might work, they might fail, they might work but for a shortened life.
Replace it with a proper LED supply.
Isolating transformer for the shaver socket.What about the other black box with the writing Custom Transformers part number 4451 ...