Sometimes but not always.
rain, hail, and wind happen.
The weather that precedes a tornado, including heavy rain and hail generally occurs in the front part of a supercell thunderstorm, with the tornado closer to the back.
Yes. If a tornado is rain wrapped rain can be drawn into the circulation.
No. Rain does not cause a tornado. However, both rain and tornadoes are caused by thunderstorms.
A tornado that is surrounded by rain is said to be rain-wrapped. Rain-wrapped tornadoes can be especially dangerous because they are difficult to see.
The Joplin, Missouri tornado of 2011 was what is known as a rain-wrapped tornado, meaning it was surrounded by heavy rain. This rain obscured the tornado from view and may have contributed to the extremely high death toll.
The tornado itself did not produce rain. But Springfield did get some rain from the system that produce the tornado.
Tornadoes are accompanied by rain, but they do not produce it themselves. Rather, the rain is a product of the thunderstorm that spawned the tornado. Hurricanes produce very heavy rain.
Usually a tornado come after rain, as most tornadoes are located in the rear portion of a supercell.
A tornado itself does not produce rain, but it can accompany a tornado. The storms the produce tornadoes, called supercells typically produce very heavy rain, often enough to prompt flash flood warnings. This rain may stop before the tornado comes, or the tornado may be rain wrapped. Some storms however, called LP (low-precipitation) supercells produce little to no rain at all, but can still produce tornadoes.
Very often you cannot see them because the tornado is obscured by the rain. This makes such "rain wrapped" tornadoes especially dangerous.
Tornadoes are usually preceded by heavy rain and sometimes large hail as well a thunder and lightning. The rain often stops before the tornado hits, but not always (in which case it is called a "rain wrapped" tornado).