What was the stand of the Mother of Believers, ʿA’ishah , in relation to the hadith of the barking dog of Haw’ab which the Prophet related to her and the decision of her to go for fight against the camp of Ali?

Your question concerning ʿA’ishah is based on lies invented by the Rafidah Shiʿah who hate and curse ʿA’ishah. You should be careful of where you take such information from. Ibn Qudamah says concerning the wives of the Prophet (ﷺ), “The best of them was Khadijah bint Khuwaylid and ʿA’ishah as-Siddiqah bint as-Siddiq who was declared innocent by Allah in His Scripture, wife of the Prophet (ﷺ) in this life and in the next life. Whoever slanders her with what Allah has declared her innocent has disbelieved in Allah, the Magnificient.”

Reference:

[The Radiance of Faith, Ibn Qudamah, pp. 126-128]