How are springs formed? Springs may be formed in any sort of rock. Small ones are found in many places. In Missouri, the largest springs are formed in limestone and dolomite in the karst topography of the Ozarks. Both dolomite and limestone fracture relatively easily. When weak carbonic acid (formed by rainwater percolating through organic matter in the soil) enters these fractures it dissolves bedrock. When it reaches a horizontal crack or a layer of non-dissolving rock such as sandstone or shale, it begins to cut sideways, forming an underground stream. As the process continues, the water hollows out more rock, eventually admitting an airspace, at which point the spring stream can be considered a cave. This process is supposed to take tens to hundreds of thousands of years to complete. ----- You should find the information on the referenced website to be helpful in understanding how springs form.
When it rains the water is ether trapped in a reservoir (lake, pond ect.), infiltrates into the ground or runs over land. When the water level in a reservoir becomes higher then the surrounding land the water finds a low point and travels down hill tell it reaches another body of water. This will eventually erode the ground and create a river channel. Spring are formed when the ground water levels rise above land and the water begins to flow freely over the land. this usaly occours near the bottom of hills or where a hill becomes steep (thats why a lot of cliffs have water running down them.
In most situations the water is from runoff after rains. The water accumulates in small creeks and then into streams and converge into rivers as you already noted. The water from springs is rarely a source for any significant tributaries to rivers. Spring water generally comes form water that has seeped into the ground nd started to flow horizontally after reaching rock that it couldn't penetrate. After traveling some distance it finds an outlet at ground level, so that water is flowing by gravitational force. Some smaller amount of ground water, in some cases, is forced to the surface by geothermal energy. That water is usually from hot springs.
I know that they definitely do not come like that so that's custom. But other than that I do not know.