Relax, the ground/earth connection is a secondary safety system, intended to protect the user in case of a fault condition such as an internal short between a live wire and exposed metal parts. Obviously I'm not advocating permanently disconnecting the ground pin and leaving it that way, but isolating it temporarily is a perfectly valid troubleshooting method. In North America, many houses built prior to the 1960s don't have any grounding at all and it's not uncommon (though not advisable) to use adapters on grounded plugs that go into a 2-prong socket. You are supposed to ground these adapters but nobody ever does. Unless a fault in the equipment exists, there should never be measurable current flowing through the ground lead, that's what neutral is for. In fact, ground and neutral tie together in the panel and come from the same place.

A GFCI/RCD works by monitoring the current flowing through the live and neutral leads, and tripping if there is a difference of more than a few mA. The assumption being that any current passing through live that is not also passing through neutral is leaking to ground, possibly via one's body.

A lot of equipment has capacitors connected between live and ground and sometimes neutral to ground to shunt electrical noise before it gets out through the power entrance. Some of this stuff (such as old Macs) were designed before GFCI/RCD breakers became commonplace, and some GFCIs are more sensitive than others. The result is precisely the sort of nuisance trips you're experiencing. You may be able to replace the suppression capacitors with some of lower capacitance, or just disconnect them entirely, accepting that you may get a bit of interference on the radio, TV, or other RF devices around the house.