None of the special Hybrid plots happen without Advanced Hybrids being enabled. The Dyson Antagonizer basically causes the Dyson Sphere to become hostile to the player, no matter what the circumstances may be. The Dyson Converter is basically a Dyson-sized matter-to-energy conversion device, and its activation is rarely good for the player.
IIRC, Advanced Hybrids triggers the Antagonizer at intensity 4 and the Converter at intensity 7, but I may be off there.
The number of Hybrid spawners and upgrade facilities is determined by the intensity of the normal Hybrids setting. Again, my memory may be off, but it's something like (intensity * 2) or (intensity * 4) Hybrid spawners per Galaxy. If you set the intensity down at 1 or 2, there won't be very many Hybrids, regardless of the level of the Advanced Hybrids setting. Intensity 4 is usually a balanced setting, but if you find Hybrids dominating your games at that level, dropping it may be a good idea.