You can make some simplifying assumptions. They will add less noise to the result than the fluctuation of line voltage this circuit will encounter, so don't worry about it too much.
Assume that the capacitor is charged at the peak of each line cycle, and then discharges thru the resistor the rest of the time. The cap therefore has to not drop more than 2 V during 17 ms. You say your acceptable output voltage is 14 to 16 volts, so assume the cap gets charged to 16 volts. Discharging from 16 to 14 volts would take .134 time constants, since .134 time constants is 17 ms, you know that a full time constant is 125 ms, which is R x C. 125 ms / 150 Ω = 830 µF.
Of course in practise you don't want to cut it that tight. At the very least, you want a 1 mF capacitor, but I'd probably use 1.5 mF or 2 mF. That will be electrolytic at this capacitance and voltage level. It should be rated for at least 20 V, although 25 V would provide longer life.
Note that the current provided by the transformer will come is large and short spikes, and their level will be directly proportional to the input line voltage. For robustness, you should design this circuit to operate properly from at least 105 V to 125 V AC. This gets tricky. There is a reason we use switchers and regulation nowadays.
Providing the copper doesn't completely totally surround the wires (forming a shorted turn) it's likely that it is an EM screen intended to reduce emissions. There will be eddy current losses but these are probably acceptable.
The pictures isn't great but it looks like there is a wire soldered to one end of the copper strip and this will further enhance the shielding quality of the copper by Earthing it on the PCB it's mounted on.
Best Answer
It is possible to provide lower voltage DC from higher voltage AC to DC subject to certain conditions. This capability is what a is generally known as a "Switching Regulator" or "Switched Mode Power Supply" (SMPS).
I'll provide an overview of a subset of the possibiliies as it gets complex" and the following should get you started. Doing a web search on the above terms will tell you much more.
This covers conversion of highr voltage DC to lower voltage DC (or rctification of high voltage AC to DC and then conversion).
Call the input voltage Vin or HV.
Call the output voltage Vout or LV.
A transformer provides isolation against shock or accidental transfer of energy from input to output or output to input in a way which is nit easily achieved by other means. Capacitors can provide isolation, with some important differences.
However, many loads cannot accept pulses of HV. If you have say 350 V DC s Vin and turn it on 1/70th of the time the effective mean Vout to a resistive load is the same as if you had had Vin/70 = 350/70 = 5V appplied. This may just possibly sound like a USB 5V power supply BUT
Electronic equipment and people do not usually like pulses of 350 VDC instead of continuous 5V DC .
The pulse of HV lead to very large current peaks and then nothing. Thermally the effect is the same as 5VDC continuousas long as you do not ry to actually make 5VDC actual.
If you apply capacitive smoothing to the pulses the current flows through resistors into the capacitor as part of the smoothing action (whether formally inserted ones or wring resistance) and you get massive energy losses. Efficiency is only lv/HV. If LV= 5V and HV = 350V you get 5/350 ~= 1.5% efficiency.
To remedy the above problems you need some sort of lossless energy store to act like an electronic flywheel to take the energy pulses, store the energy in the "flywheel" and then release it across the rest of the cycle when the switch or chopper is off.
The only electronic element able to do the above is an inductor. There are electromechanical means but they prove to be far less practical than an inductor.
An inductor looks like one winding of a transformer but is essentially far simpler and the basic method of its use is different. In a transformer the majority of the energy transferred is NOT stored in the inductors. In an inductive smmothing system it is.
When a voltage pulse of HV is applied to an inductor the current ioncreases approximately linearly and when the HV is removed the current decreases approximately linearly as energy is given up to the LV circuit.
The resultant circuit is what is known as a "buck converter". It still uses an inductove element but not a transformer. The switch can operate at high frequency allowing the inductor to be far smaller and lighter than a mains based transformer. There is no isolation of input and output.
This circuit shows a basic buck regulator. When the chopper switch is closed energy flows via the inductor to outpit. When the switch is open enrgy continues to flow from th sored emergy in the inductor. The diode provides a path for the energy flow.
A simple practical buck regukator can look like this. The switcyh is inside the IC bewteen Vin and Out.The connection to FB allows the IC to adjust the chopper on/off ratio to get the correct Vout for varying load etc.