Re: Toilet leak
Probably 99% of the time the problem described is caused by and solved by replacing the flapper. Flappers are not that expensive and don't require a plumber to replace them.
Going through the test of shutting off the water to the tank on an old toilet with a shut off that may not have been used for years could create a lot of other problems including plugging of the fill valve, blockage in the fill line, or the shut off it' self leaking.
To me the logical thing to do is to replace the cheap flapper which causes this problem 99% of the time. Then if it doesn't take care of the problem start looking for the other exotic causes.
Be sure you live your life, because you are a long time dead.-Scottish Proverb