Can a load on the line side ever trip a GFI?
Although in theory a device on the line side of the GFCI will not trip the device, it can in practice.

In theory, a GFCI will only trip when it detects the current is not balanced on its load terminals. Unfortunately, the design of a GFCI makes them susceptible to interference. They can pick up electromagnetic signals, and also may also trip with wild swings of line voltage. There was a recent presentation at DEFCON that demonstrated how a (strong) radio transmission can cause the GFCI to trip, and even self-destruct.

I have personally experienced unwanted GFCI tripping from motors in my home: an exhaust fan and a garbage disposal. The GFCI tripped as the motor was turned off. In these instances, they were older motors and the circuit did not carry a safety ground (two-conductor NM cable).