Do you also do pushing from the side?
Why do you think hanmi has to be "stable" to the rear? + What is best for being pushed from / for pushing yourself to the front?
Do you try to move from hanmi having your axis on different positions? Do you feell differences?
Do you investigate the connection / the "way" back leg -> center / tanden -> front leg and backward? And if so: Does it feel different where you have your leg?
Yes, if you want to test stability, side pushes are useful, too.
If a stance is weighted forward, then it's easier to be drawn off balance (and overextended) forward. If weighted backward, then it's easier to be overwhelmed by a strong attack. A neutral stance isn't vulnerable to either.
If your weight is on one foot or the other, that foot will be more difficult to move.
I'm not sure I understand your last question.