Is a mother's rejection of marriage to a divorced woman—out of fear of what people might say—considered an impediment that justifies not proposing to her, and how does one performing Istikhara ensure that what they are about to undertake is good?

Obedience to the mother takes precedence over marrying a specific woman, especially if her refusal is based on an acceptable reason, and not merely on whim. The mother's initial refusal is not considered a sign of difficulty after performing Istikhara; you can try to persuade her again with gentleness and affection, and by involving someone who can convince her. If she insists on her refusal, then you must turn away from this woman in obedience to your mother. The expansion of one's chest and the facilitation of affairs after Istikhara are signs that the matter is good.