The underlying theory of blood type diets is that people with different blood types digest lectins differently, and that if people eat food that is not compatible with their blood type, they will experience many health problems.
Leptin is a potent stimulator of the immune system. Leptin receptors on monocytes cause the release intracellularly of STAT3 which cause the monocytes to release IL-6, IL-12 and TNF alpha. The IL-6 goes to the liver and increases production of CRP.
Leptin does not influence fat storage levels; rather, leptin is secreted by fat stores. You are right in that leptin deactivates hunger. It does so by inactivating NPY & AgRP (hunger) neurons, and activating POMC (satiety) neurons. Obese people have high levels of leptin because they have a lot of fat.