A raindrop forms at the temperature of the clouds, which is always cooler than the ground below because of the cooling effect that lower air pressure and higher altitude has (this is why it snows in the mountains). As a consequence, a raindrop will always be cooler than the air beneath it that it falls into. Nonetheless, if the entire air column is warm, then the raindrop can be pretty warm, too, and water holds a lot more heat than air does, so getting splashed by warm water can feel a lot warmer than being hit with a burst of warm air.

That's an interesting question, about the rain drop.In tropical climates, the rain doesn't feel very cold, but I don't expect the rain drop can get very warm. Here's my thinking:
1. when the rain drop falls, the air flow past it will cool the rain drop down.
2. When it's hot and damp, I think the rain gets sort of misty because of evaporation.
3. Evaporation - going from liquid to gas - takes LOTS of energy. This removes a LOT of heat from the liquid, which is a rain drop in this case.

I found scientific articles on 'warm rain,' but that's just rain that comes from a cloud whose temperature is above freezing everywhere, which doesn't sound very warm to me.

I don't have any numbers to give you about rain temperatures, but I expect there's lots of interesting info about your question somewhere on the Internet.

Yes. In temperate climates, rain usually starts out as tiny ice crystals in clouds high in the atmosphere (where it is much colder than at the surface), so these raindrops feel cold. On the other hand, warm rain that forms lower in the atmosphere (where the ice crystals do not form) is common in the tropics where warm moist air moves upward by convection.