Assuming the replacement screen assembly and board are known-good (have you seen them both working with your own eyes?), you've ruled out pretty much all the hardware. Yes, the inverter and inverter cable are part of the display assembly. A1181 inverters just about never go bad, but the cables do fairly often. The screens go out even more often than the inverter cable.
I am not 100% sure from the description of your laptop, but judging from the symptoms I'm willing to bet you have an 820-2279-A board (the number is printed under the right RAM module) with a 3-wall inverter cable (flat and thin connector, rather than the big and chunky 4-wall that plugs in from the top). 3-wall means the socket on the board has 3 surrounding edges and is open on one edge, whereas the 4-wall socket has plastic edges all the way around, and so you plug in from the top.
Anyway, the 3-wall cable has 4 wires that go into tiny holes on the connector at the end of the cable, which in turn goes into the socket on the board. The 4 wires are notorious for coming just slightly loose and causing the behavior you are witnessing. They are not necessarily even disconnected from the plastic connector, but if you look carefully you can see through the tiny slits on the connector that they may not be fully seated. While it is connected to the board, you can use a good pair of tweezers to grab each wire and push it all the way in.
Also, the fact that the motion of moving the screen hinge is apparently tugging on the inverter cable suggests that you may have screwed down the left screen hinge onto the inverter cable improperly. You may want to remove the left screen hinge bracket (the piece that has 3 screws in it), and see if you can adjust the inverter cable a little, and give it some more slack.
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