This is a fun question. I feel your plight and frustration. The answer is multi-point.
The TLDR version: Yes, a restrictive component upstream from your faucet control and the spicket can introduce the conditions needed for a leak between those two points.
When the faucet itself was off however, all of the water pressure in the home in the pipe is pushing on all sides of the pipe's and faucet's inner surfaces. Home water pressures can be about 60 psi (different codes in different places), but this PSI is regulated by a regulator on your main water supply. All the water in your home is restricted to whatever this is set at.
Now, when you open a faucet, you are giving that water a place to go. Think path of least resistance. The water goes there, but unless the pipes it is going through are too small (like a refrigerator water line) that water will come out at 60psi.
Suffice it to say for the example, the pipe between your shower handle and the shower head can handle say 40psi…. when you open the faucet, with no head, all 40 psi (nearly all), blasts out into the tub. Now put your thumb on it, and some MORE pressure pushes on the pipe sidewalls, while less than 40 comes out. Now put a low flow head on it, and even less.
So yes — if a pipe was weak to begin with, it can start leaking. Now, as far as the cold supply leaking because of the new head? BALONEY. Because when the water is OFF, it is seeing all of the pressure at those connections.
What is most likely the issue with the leak and the faulty faucet (I assert…) is that the act of you touching the shower head pipe had more vibrational impact on the overall piping installation. Also, calcium or other water impurities built up in the pipes may have broken free. This could be what caused the water mixer to "seize up" and also could be what weakened the pipes more than they were before you started the task.
How old is the house? These things do happen. Also, the plumber (while not at any fault), also jostled these connections, and he is right, if the job was lacking in quality, then everything you experienced could happen.
Best Answer
I've typically seen faucet handles configured in one of two ways. For radially symmetric "knob" type handles, both may be set up so that the behave like common screws - righty tighty (off), lefty loosey (on). For "lever" type taps like the one linked in the OP's comments, it may be more common to have both handles rotate "in" toward the user to turn the tap on - the off position is at 9 and 3 o'clock for the left and right taps, while the on position is at 6 o'clock for both. If the faucet taps have long handles or are close to the wall, it's not even possible to have the levers rotate away from the user to 12 o'clock, but they can always rotate toward the user to 6 o'clock.
Which way you want the taps set up will be a matter of personal preference, handle type, as well as the physical realities of if it's even possible to turn the tap backwards toward the wall.