Memorizing the 99 names of Allah

Q: What is meant by "memorizes" or "preserves" the 99 names of Allah in the Hadith below:

عَنْ أَبِي هُرَيْرَةَ عَنِ النَّبِيِّ صلى الله عليه وسلم قَالَ لِلَّهِ تِسْعَةٌ وَتِسْعُونَ اسْمًا مَنْ حَفِظَهَا دَخَلَ الْجَنَّةَ

Cycling

Q: I am an active cyclist. Crashes are often and we tend to get a lot of cuts and bruises on our body which causes our hair to get stuck to it and infect it. Is it permissible for a man to remove his legs and arm hair in this situation?

Washing a musalla that was touched by a shoe

Q: My mother uses a sandle which she takes to the toilet also. A few times she accidentally stepped on my musallah. I touched the part and it was not wet. With my hand only, I clean the dust without using any water as I am not sure where she exactly stepped on. She also stepped on the musallah with her shoe which she wears when she goes outside. Do I have to wash the musallah? I am currently reading salaah on the same musallah. I have doubts about another musallahs because my nephew urinated on them and the house worker washed them. I dont know if she washed it thrice or not. Can I read salaah on them as she used a detergent?

Repenting from homosexuality

Q: My best friend commited a major sin, he committed homosexuality. I told him that it’s a very big sin and Allah punished the people of Lut (alaihis salaam) for this sin. I told him that he needs to repent and turn back to Allah. My question is, does Allah forgive homosexuality?