What is the ruling on the prayer of someone who has prayed for many years after performing a cleansing or cooling ghusl without ablution, and without the intention of removing ritual impurity, knowing that he might have touched his private parts during the ghusl, and was ignorant that touching the private parts invalidates ablution? Does he have to repeat those prayers, or can he adopt Ibn Taymiyyah's fatwa that an ignorant person is not required to make up prayers? And does Ibn Taymiyyah's fatwa regarding the validity of praying alone at home apply to him?

A cleansing ghusl (bath) does not remove minor ritual impurity (hadath asghar). However, the Hanafis hold that it removes both types of impurity (minor and major), and they do not stipulate intention (niyyah) for purification, nor order (tartib) in ablution (wudu'), nor do they consider touching the male organ to nullify ablution. Therefore, your prayer is valid according to their madhhab. Rinsing the mouth (madmadah) and sniffing water into the nose (istinshaq) are Sunnah in wudu' according to the majority of scholars. It is permissible to act upon a less preferred opinion (madhhab marjooh) after an action has occurred and it is difficult to rectify it. Therefore, you have leeway to adopt the Hanafi madhhab in this matter. Ignore obsessive doubts (waswas) and do not give in to them.