Yes, ballcock valves can fail over time due to wear and tear, mineral build-up, or low-quality materials. It's quite common for them to stop shutting off properly, which leads to overflow problems like the one you're experiencing. Adjusting the float rod as you did is a temporary fix, but...
You mentioned the possibility of the u-bend syphoning out and running dry, which could allow sewer gases to escape into your utility room. This is indeed a possibility, especially if the stand-pipe is not installed correctly or if there's insufficient air gap between the stand-pipe and the...
Typically, the water shut-off valves are located under the sink, near the pipes that connect to the tap. They're often small valves with a lever or knob that you can turn to shut off the water supply. However, the main shut-off valve is usually located where the water supply enters your home...