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.
Not really, the key is to be directed in the application of heat to the required joint, Almost certainly you will melt the solder on the adjacent joint, however the solder should re-set quick enough. Bigger concern is to make sure you don't apply too much pressure to the fixture and dislodge the loose joint (at heat) --
Remember, the solder should not be contaminated and should re-set well enough.. the nice thing is that its pretty easy to test (i.e when you turn the tap back on) - If you are concerned, or there is a leak, you can simply add more heat and apply some solder with new flux.
Best Answer
I don't see any reason why unintentional reflow of the solder on that nearby elbow would inherently be any problem. Try to minimize disturbing it while the solder is molten -- don't pull the pipe out of the elbow, and hold it with pliers so the pipe won't rotate while you're trying to work the other fitting off the same stub of pipe.
It'll probably work out just fine, and if you have the parts and tools on hand you'll be ready to deal with any trouble that might arise.
The biggest thing is to make sure you keep all the water cleared out while trying to solder. Have a dissolvable sponge (such as a slice of white bread) on hand to stuff up into the pipe to hold water at bay, just in case.