Is it permissible to practice aerobic dance exercises without music while recalling or mentally humming songs and steps, and what is the ruling on music in Islam, taking into consideration the opinions of some فقهاء (jurists) and صحابة (Companions)?

We praise Allah for your Hajj and ask Him to accept it. Your actions of abandoning music and dancing are blessings of obedience. Repentance nullifies what came before it, so thank Allah for that. As for your first question: Aerobics accompanied by music is forbidden (haram), and the prohibition is greater for those who teach it to people. If it is free of music, performed with the 'awrah covered, and without mixing of genders, then there is no objection to practicing it, as long as there is no fear of returning to the forbidden. What reminds one of sin must be avoided. As for the second question: The proofs for the prohibition of singing are numerous. It has been narrated from the Companions and the four Imams, and some of them have even reported a consensus. Those who disagree are few. And those who permit it agree on its prohibition if it involves mixing of genders or obscenity, as is the case today. Scholars have refuted the arguments of those who permit it.