Electrican thinks that if the fan plug is not perfectly straight the plug may not be touching the ground right. The plug on the fan looks pretty darn straight to me. Suggestions?
Your electrician doesn't seem to understand how the GFCI works. The devise does not need a safety ground (green wire) to operate. The GFCI senses the difference in the current flow between the hot and neutral wires. If that difference is more than 5 milliamps then the GFCI is tripped and both the hot and neutral circuits are opened.
The assumption is that the only thing that could cause a difference in current flow between the hot and neutral, would be a short to ground with you or me being the conductor. The GFCI trips so fast that the current doesn't have enough time to kill us.
It's kind of interesting how the GFCI senses the current difference between the hot and neutral wires. Well, I at least I find it interesting... feel free to fast forward.
The GFCI has a tiny coil with the hot and neutral wires passing through it. It is exactly the same way clamp multi-meters work. If you have ever tried measuring AC current with the clamp meter you know that if you put the clamp around both hot and neutral you will not get a reading on the meter, because the current flows are equal and cancel each other out. You have to pull the wires apart and clamp either the hot or neutral to get a reading. So, if you were getting some current reading while clamped around both wires, there is a fault to ground somewhere or there could be a break in the neutral wire. When the GFCI coil sees current above 5 milliamps it assumes the worst and trips the circuit..
Like others have said, there several things that can trip the sensitive GFCI that are not a safety concern.