Okay. Although I can't be 100% sure from my keyboard across the Internet, I'm going to assume everything is assembled as it should be without physical damage (visible or otherwise) inside the case.
[begin officially published by Apple - and recommended - hardware troubleshooting answer]
If you think you may have a hardware issue, you can check it out first with the built in utility on your mac:
Using Apple Hardware Test (instructions copied direct from Apple):
1) Disconnect all external devices except the keyboard, mouse, display, and speakers. If you have an Ethernet cable or external DVD drive, disconnect it.
2) Restart the Mac, hold down D while the Mac restarts.
3) After the Mac restarts, you should see the Apple Hardware Test chooser screen. If you do not, Apple Hardware Test may not be available on your Mac. You may be able to start Apple Hardware Test from the Internet.
Reconnect your Mac to the Ethernet network, then restart your Mac and hold down Option-D.
4) When the Apple Hardware Test chooser screen appears, select the language you want to use, then press the Return key or click the right arrow button.
5) After approximately 45 seconds, the Apple Hardware Test main screen will appear. Follow the onscreen instructions.
If Apple Hardware Test detects a problem, it displays an error. Make a note of the error. Then check here:
https://support.apple.com/en-us/HT203747 to look up the error code to point you in a direction of what may be wrong, and a possible solution.
If that doesn't get you anywhere, taking the machine to an Apple store (or authorized service center) would allow the techs to use additional diagnostic tools to look deeper than the AHT can go to get to the issue.
[end official answer]
Without being able to actually look at it and see what is going on, that question mark generally means for whatever reason, the machine can't find an OS to boot from.
Considering the mac can boot into recovery mode suggests at least that likely the internals are working. Seeing that the new SSD (known good drive) crashes during OS install suggests that you might be on the right path, assuming nothing else has changed and the problem isn't further upstream. A simple way to troubleshoot that cable is to put the SSD into an external drive enclosure (or use an adapter cable made for just this purpose - best $30 I've spent on computer tools in a long time) and then try your OS install to the new "external" drive. If that works, then you know you've got something going on either with the cable or the port on the board the cable is connected to. If you can do a web restore, you can do that to a big enough USB drive, if you have one handy instead of your SSD if you don't have an enclosure or adapters to get that going.
There are a few more long shot possibilities, but I wouldn't want to get you into more trouble throwing out little more than guesses from across my keyboard that could do more harm than good without being able to own the repair from end to end.
