the green LED comes on. The screen lights up for 5 seconds as normal, then goes black again. The green LED remains on.

There is no standby light. Who asked about this? (According to page #20 of the Operating Instructions, your set DOES have a standby light. It is one LED to the LEFT of the Power LED.)
When power is turned on the power led is solid. (The Power LED & Relay both stay energized until the set is 1) TURNED OFF? &/or 2) UNPLUGGED?)
The screen shows a bar @ the top with input and channel (Does Input that's identified have an active source connected to it? AND IS Cable/Antenna jack connected to a Antenna or Cable system? AND Is channel identified an active channel on the antenna or cable system?)
and no signal box @ the bottom, but only for a second or two and then goes away. When is "Sound but no picture"? Are we talking a relay click or actual intelligent audio?
The screen doesn't stay up long enough to select an input.
I have changed out the backlight inverter board but it still does the same thing. ("IMPORTANT: If a unit is found to be turning on with no backlights, yet it is not shutting down (power LED remains steady green), do not replace the LCD panel or inverter board. There cannot be a defective backlight or inverter board without a 6-blink shutdown.")
I hope this information is helpful and thanks for any help you can give.