What could make an LCD stop working permanently after being connected for 5 seconds to Arduino?
It's unlikely that you have killed the unit as this is pretty hard to do from the data interface and your picture is typical for an LCD that has not gone through the initialization sequence.  A dead LCD driver will usually display nothing and what your picture shows is that the LCD controller itself is operating.  If anything is dead, I would suspect the I2C interface board as this would yield the same result you are seeing.  It may also be that the timing of the I2C interface is right on the edge which may explain why a long cable'd setup is working.

Note that if the initialization is not done, no other commands will work.  Note also that the initialization timing and sequence of events is ABSOLUTELY CRITICAL and missing any single step or time requirement will give the result you are showing.

Starting at page 10 of this instructional PDF is my favorite and the basis of my driver that I have been using since the 80's.



The driver I use works on every display I have ever hooked to it, regardless of changes in the particular project's processor clock speed because I paid strict attention to the worst case timing requirements.  If the timing of the I2C interface violates one LCD it may not violate another, i.e. one display can tolerate bad timing while another can't.  There are also issues if the LCD driver uses the busy flag or not:  Blind data transmission without reading the LCD's busy state will work if the LCD is fast enough and the driver is slow enough, otherwise 'No bueno!' plus the interface is not going to operate at its maximum speed.  When using the busy flag (Bit 7 of the interface) there is solid handshaking which confirms (Instead of "Trusts") that the LCD is ready for the next cycle and allows operation right up to the limit of performance.  

What I would suggest is that you put the interface on a logic analyzer and verify that the timing is not being violated.

Timing tolerances and variation between parts is like lug nuts on a car... If you car has 5 lug nuts per wheel, it will probably work with 3 or even 2 but NOT UNDER ALL CONDITIONS.  SOME LCDs will be happy with bad timing but ALL LCDs NEVER will be.