The "open ground" indication lights because the generator's windings are isolated from the ground pin, which is tied to the generator's frame. The frame is not grounded unless you connect it to a solid ground, like a grounding rod. Normal commercial power has a hot and neutral line, with the neutral side tied to ground potential. Since the generator's windings are isolated from ground there is no need to ground the generator frame.
GFCI outlets will operate with normal protection, as they operate by sensing a difference in current between the so-called hot and neutral lines. Due to the isolated windings, even if a fault to ground occurs there should be no shock hazard from current flow to ground. If something fails inside the generator that causes the windings to lose their isolation to ground, the GFCI will operate normally.
