Hi Jane, ball of foot only on stair edge so mid part of foot is off.
Having more of the foot on limits range of motion at the ankle and the stretch you're feeling is probably more of the lower calf muscle, when you want to be stretching both ideally. To target both calf muscles do the heel drops with a straight and bent leg. E.g. 15 x 3 straight leg, then same with bent.
If stretching before running helps you then do it. I do motion stretches before, so no holding, just taking the joints through their normal ROM as part of my warm up.
Best stretches to do after exercise are static ones, where you hold for 20-30 secs. As for what muscles - all the main ones: calves, hams, quads, hip flexors, lower back. Having said that there's not one jot of evidence that stretching itself prevents injury.
Some runners actually benefit from not being too flexible as the muscles retain more power than lax ones. For injury though you want a nice compliant musculotendinous unit for it to function properly, as scar tissue hasn't the same properties as muscle tissue.