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?
Dunno about the HAI Electric Strike, but yes, the inrush current may be 2 or 10 or 40 times the hold current. 4.5A rather than 450mA.
Not because the coil current is greater when the plunger is outside the coil, as is the case for AC solenoids.
Rather, because the hold-in current is artificially reduced, either by using a separate hold-in coil, or by switching a current limit into the circuit. You do this to get a good pull-in (what you are not getting) but not wasting power and burning out the coil when the door is left unlocked.
A simple tranformer plug pack had no effective limit to current supply: if you shorted it out for long enough, it burned out. A modern switching power supply does have an inherent current limit: if it doesn't burn out it will still only transfer the rated current.
Best Answer
Use your signal generator set at 50 Hz or 60 Hz and, through the primary hole wind ten turns (or more) but note the number. Drive current through those ten turns and monitor that current using your multimeter. If you can get 100 mA RMS through 20 turns, that's equivalent to 2 A RMS in a single primary wire. If that isn't enough, wind more primary turns (say 100) or, build a little DC power amplifier to produce more current into the home-spun winding.