Constantly running toilet with floatless fill valve
The internal gasket/diaphragm needs replacing, check out this link.

It does appear that you could extend the flush valve overflow tube as the height appears to be below the water level mark on your tank. That mark is actually the right level for optimal flush performance for your toilet. If the valve is worn out it may not help though, although the weight of the additional water may get it to stop. You might try opening the valve up and gently rinsing the gasket, in case there is debris or build-up interfering with proper operation (turn the water supply off first!)