By shower elbow, you mean this part, right?
If so, then to remove any remaining part left in the copper, get yourself some LONG needle nose pliars
Use those to remove any remaining plastic from the copper.
You can buy a replacement at pretty much any Home Depot or Lowe's - so buy one AND some teflon tape
IMPORTANT DO not buy the yellow package - that is for GAS, not water. Blue and white or red and white packaging only!!!
Wrap 2 or 3 layers of tape around the wall end of the elbow. Wrap it so that the tape follows the threading - so hold the pipe with the threads near you, start near you and wrap CLOCKWISE around the pipe. That way as you screw in the pipe it won't peel off the tape. Not too much tape or you won't get the pipe to fit!
Insert the elbow and turn it COUNTER CLOCKWISE slowly at first, until you feel the threads kind of click into place (avoid cross threading) and then turn it clockwise. Hand tighten only but make sure it's nice and snug and vertical.
Then reinstall your shower head per mfg instructions.
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
It does look as though someone made their own bent arm from copper pipe and sweated it onto a Male Adaptor (MA), you can see the hex shoulder of the MA inside where it threads into the right angle fitting. But in doing that, they created a weak point in that the copper pipe will likely collapse and twist off before that threaded joint breaks free. The MA has that hex shoulder to use to tighten and loosen it BECAUSE the copper pipe is too weak for that, but now that hex shoulder is buried into your wall to where you may need to cut out the wallboard to get access to it. That's a shame.
I would first try putting some penetrating oil on that fitting and tap the pipe for a few minutes to work it in, see if that helps break that fitting free without having to open up the wall.