Based on your troubleshooting, which was well done, the first thing to check is the condition of the connectors on your logic board. They may have been damaged from the repeated screen changes. Most people don't realize this but those connectors are not rated for many connect/disconnections (typically a few dozen). Most of the time you can do it over and over again but they are very delicate.
If the connectors are in good condition, check to see if any surrounding components were accidentally removed. If that is okay, then I suspect that you damaged the backlight circuit on the logic board when you replaced the screen. Did you disconnect the battery before removing and installing the screen assembly?
You could probably do this 50 times and not damage the backlight circuit but it does happen and we see this "problem" pretty regularly on iFixit.
If that is your problem (or a damaged connector), you will need to find a repair shop that does micro-soldering repairs.
Yeah, I think I mis-read your question. Its difficult sometimes to keep track of multiple phones and components ;>).
Anyway, if I understand correctly, the original phone needed to be repaired for something (cracked screen?) and while the new screen did not work, you tried it on a spare phone and it did not work there either. Then the original phone blew up during a secondary repair effort (battery?).
So now you only have the spare phone, its original screen and the new replacement screen. But now neither screen works although the phone is detected by iTunes and appears to charge...correct?
So if this is the case, you should still check the condition of the connectors. It's unlikely to be the problem but you have to eliminate that. Then my question remains, did you disconnect the battery before doing all of this screen swapping? That is the most common cause of backlight circuit failures and you haven't answered that yet.
Assuming everything is in good condition and the battery was properly disconnected, then it is indeed possible for a poor quality replacement screen to have damaged the backlight circuit on your logic board(s) by drawing excess current. The part that fails the most is the filter (a ferrite bead). In some cases, more extensive damage can occur to the rest of the components responsible for driving the backlight.