IMPORTANT: (1) Shorting one winding of a transformer that is hard coupled magnetically will place a short on that other winding as well. You will get magic smoke as shown.
(2) Arduino as shown floats wrt TRIAC with only 1 lead connected. you MUST NOT rely on people assuming what connection you meant - assumotions will vary.
(3) Touch the Arduino and die.
It makes little sense to provide an isolated Arduiono supply and then to throw away the isolation, even if isolation was not your main aim.
(4) Having a mains load that has a designed small current flow through it when off is lethally dangerous and also illegal in all sensible administrations.
(5) Your desire -
And since I want my design working exactly as a commercial device this is the ONLY way I want to connect.
is doable but you have to do it properly. The mechanical thermostat is effectively an amplifier with an external energy source - it uses thermal energy to move a bimetal strip . This is incredibly efficint anergy wise - the fact that you are competing with an in many ways superb mechanical solution in the form of a mechanical differential to switched mains comparator is something you have to realise and deal with. Doing so by getting your powering energy via the serial leads in unacceptable.
Potential (pun noticed) solutions include a local battery charged by voltage drop when on (about 1.4 Volts needed) or a FET (so about zero drive power and cap or supercap storage or an energy harvesting solution.
The fact that this is "easy" mechanically and "hard" electronically is your problem, not nature's. Mother nature makes the rules and we are obliged to obey them.
(6) Using a ruler and a bit of preplanning when drawing your diagram (so eg the diode bridge is not jammed into a tiny space) can get you an almost Olin proof drawing. Almost. As Olin would point out, you are asking people to assist you. Giving them a modicum of respect by providing a diagram which is clear and easily readable is liable to be at least a good idea. Hand drawn can be fine (I say, some may disagree) but make it tidy.
There are ways of doing what you want - either with two wires and external power of some sort, or by running a "neutral wire, as you note. We can discuss these, but first address the above so we know what path to take.
160 mA is high. What's that used for?
Brushed universal motors are largely independent of AC frequency, and as you have heard, will also run on DC. Their maximum speed is normally way above any synchronous speed referenced to line frequency. In their construction, the phase angle between the rotor and stator fields is set by geometry, not line phase. This is also why they run at high speeds - they just don't care about the line frequency. This is ideal for appliances such as vacuum cleaners and tools such as routers and planers, which don't need to maintain constant speed under varying loads. On the down side, they are typically not very efficient at low speeds, but for an application where they can run fast, that's not a problem.
As long as you get the voltage right, you'll be fine.
Variable speed controllers will also work well.
Best Answer
Equipment designed to work with one frequency power will not necessarily work with another frequency.
The combination of 50Hz and 60Hz would cause generators to fight eachother