Adding: a thin piece of water soluble stabiliser tacked behind the hole makes the fencing much easier on fine gauge, then it rinses away at the end.
Wash before you mend, not after. Wool relaxes and shifts in the wash and a mend done on unwashed fabric can end up puckered once everything settles. Hand wash cool, roll in a towel, dry flat, then repair.
Almost certainly tension plus yarn weight. Darning yarn is often thicker than the sock yarn, and when you add a stiff woven grid on top of a stretchy knit, you get a rigid patch inside a flexible fabric, which is exactly what rubs. Two fixes that work together: use a finer yarn, ideally a single strand of sock yarn or embroidery floss split down to two strands, and leave your rows visibly slack so the weave sits at the same tension as the knit around it. It looks messier off the foot and feels like nothing on it.
Do a test one and put your finger inside. If the patch resists bending over your knuckle, it's too tight for a heel.
Whatever you pick, wash and dry the jeans the way she will before you measure. Four inches is a lot to cut off a denim that might still shrink half an inch.
A little firmer is normal and it does soften, mostly over the first two or three washes as the fibres bloom and settle. What isn't normal is a patch that won't bend with the fabric. The test is to drape the area over your fist: if the mend curves with everything else, leave it alone, it'll be fine. If it stays flat like a coin while the fabric bends around it, it's too dense and it will wear a hole right at the boundary within a season.
Good catch. Extending now is ten minutes, doing it after the halo blows out is a much bigger repair.