2 AC to DC converters in series problem
@Christos, 5 requires a live and neutral connection from the mains. 

With your poor understanding of electrical circuits at the moment you should not be doing any mains wiring. Stick to low voltages. Use plug-in power supplies and only plug them into sockets - not the lighting circuits.

[Edit after Christos' update.]
There are a few problems but let's make an easy example. Let's say you have a 100 W lamp and a 5 W Arduino power supply and let's make the maths easy by saying the mains voltage is about 200 V. The bulb would draw 100W / 200V = 0.5 A on full mains voltage. The PSU would draw 5W / 200V = 0.025 A. Since they're in series the PSU will limit the current and the lamp may glow a little but certainly won't light up.

Your circuit doesn't show a dimmer element in series with the light bulb so it's hard to know how you're hoping to control it.