Is a wife's Umrah valid if she entered ihram while experiencing some spotting, and then the bleeding increased, so she performed ghusl, took precautions, and completed her Umrah, knowing that she is breastfeeding and has not had her menstrual period for six months, and believes the blood is due to fatigue and not menstruation?

The default assumption for blood discharged by a woman is that it is menstrual blood. Therefore, your wife must enter into ihram from the miqat and not pray or circumambulate until her situation becomes clear. If the bleeding continues for a day and a night as is usual, then it is menstruation, and her ritual bath (ghusl) for it is not valid until she sees purity. If it stops in less time than that, then it is irregular bleeding (istihadah), and she must make up the prayers she missed, then purify herself and complete her Umrah. If she performed ghusl before achieving purity, her ghusl is not considered valid, and she must repeat her purification and Umrah. As for the prohibitions she committed: if they were for luxury, there is no expiation upon her. However, if they involved destruction, such as shaving hair or clipping nails, she must offer a fidyah (expiation) for each prohibition.