What did you use to connect your tub spout?
I plumbed my bathroom with Pex and was heavily cautioned NOT to use Pex to run from the main valve to the tub spout because it would create exactly the issue your describing.
Since the inner diameter of Pex is slightly less than regular copper (or threaded brass fittings, which is what I used) it will create a slight flow restriction to the tub spout and (inadvertently) divert it to the shower head.
Just a thought...
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
If you replaced the cartridge a set it exactly the same way, you may have set it to the wrong "off" position by duplication of the "leaking" one. The temperature limiter also sets the stop point on many diverter.
See "stop tube kit"
soilco.co
Test
To test if this is the issue, remove the hardware and very slowly turn the cartridge and try to find off dead center.
If this fails
If this doesn't solve the problem then it's likely that the diverter body has worn or has developed sediment. It may need to be thoroughly cleaned or replaced.