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The rain is likely interfering with the pollination process for the beans. The wet and humidity makes the pollen clump up and it has a harder time transferring within the flower so that pollination can happen. Without pollination, pods can not form.

Try giving your bean plants a gentle shake every few days. This will help them distribute their pollen a little better and should get you some bean pods.

Poor pollination may be to blame. Here is more information: https://www.gardeningknowhow.com/garden-how-to/beneficial/insect-pollination-process.htm

It sounds like you may have nitrogen heavy soil. All beans are nitrogen fixing plants so be careful to avoid heavy nitrogen feedings, as nitrogen-rich soils result in excessive vine growth and little to no beans. This article should also help: https://www.gardeningknowhow.com/edible/vegetables/beans/bean-blossoms-no-pods.htm

Try adding some phosphorus to the soil. Bone meal is a good source of this. Here is more information: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm

Butter beans are normally spaced anywhere from 6-10 inches apart, so I would thin them down to within this range.

It depends on the variety. Some are bush and some are pole.

This depends on the variety you are growing. Some types are climbing while others are bush-like.