Is it permissible for a menstruating woman who has exited the state of Ihram by stoning Jamrat Al-Aqaba and shortening her hair to go with her brother and his wife to perform Sa'i, and is it permissible for her to wear the niqab during Tawaf Al-Ifadah and Sa'i?

The default is for you to wait until you are pure to perform the Farewell Tawaf in a state of purity. Your brother should wait for you, or leave you in a safe place and return to pick you up if your accommodation is within a travel distance (qasr), or he can take you with him and you return after your purity to perform Tawaf while remaining in the state of ihram for other prohibitions. If this is not possible, what needs to be done has been explained in Fatwa no. 33304.

Since you have performed the stoning (ramy) and shortened your hair (taqseer), you have completed the first stage of disengagement from ihram (tahallul asghar), which permits a woman to cover her face with a niqab and wear gloves.