Is a mother suffering from epilepsy, high blood pressure, and neurological diseases—whose condition is affected by crowds and walking—excused from performing the Hajj obligation if there is a fear of her being exposed to danger because of it? And would performing Umrah every two or three years suffice for her instead of Hajj?

If the mother does not possess money for Hajj, then Hajj is not obligatory upon her, nor is the father obligated to arrange Hajj for her, nor is it obligatory to appoint someone to perform it on her behalf. Your trip with her for Umrah is an act of righteousness towards her.

However, if she possesses money, but traveling for Hajj causes her extreme hardship: If her illness is expected to subside, she should wait until she recovers, then perform Hajj. If her illness is not expected to subside, it is obligatory for her to appoint someone to perform Hajj on her behalf, provided that the proxy has already performed Hajj for himself.