Some researches in China showing that "ethylene" is responsible for some of premature flower bud falling. They use STS 1% solution (used in film development which contains silver ions), an anti-ethylene agent to spray potted tree peonies, the treatment reduces the bud falling by 55%.
STS is poisonous to the environment, so I don't think you would want to use it on your plants. But if you can find other anti-ethylene agent or provide good air circulation around your plants, it might reduce the ethylene gas build up.

With respect to pinching side buds - that mainly allows the plant to focus energy on the main buds to give them bigger blooms. It's not really required to pinch unless that's what you want.

As for the bud rots - the most common is "bud blast" caused by botrytis, which often manifests itself in cool wet springs if the fungus is in the area. If your plants show symptoms (buds abort and stems appear to be dying during the growing season), then you want to remove and dispose of the infected stems in the garbage (and clean any pruning shears that you use to cut them). Also make sure you do a good stem/foliage cleanup of the peonies in fall so that the fungus has less chance to overwinter in the soil near the plants. You can also try anti-fungal products that target this disease, which are generally sold at nurseries, etc.

Another cause of bud rot is that some cultivars have open tops on the buds if it is rainy or wet the water will enter the bud and cause it to either abort or just rot the flowers from the inside. In this case only a different cultivar is the answer.