Yes, the Earth's magnetic field has reversed in the past and may do so again. These reversals take place over thousands of years. When the field is undergoing a reversal it reduces in magnitude and points towards the equator initially, then increases with the opposite polarity (what was once the north magnetic pole becomes the south pole, and vice versa.)
Space storms would be very much affected by the field reversals at Earth. Because during the reversal Earth's magnetic field would be rather small, and point towards the equator rather than the poles, the Sun's energy would be able to penetrate closer to the atmosphere, because Earth would lack a strong magnetic field to shield it.
Energetic particles streaming along magnetic fields connected directly to the Sun could come inside Earth's atmosphere and increase hazardous radiation. Storms and substorms might still happen, but the direct solar wind impact onto the upper atmosphere and the penetration of solar energetic particles directly into the atmosphere (without the protective effects from Earth's magnetic field) would dominate.
Once the field reversed completely, phenomena at Earth (including storms and substorms) would be similar to now.