All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, may Allaah reward you for being keen on calling non-Muslims to Islam and we ask Him to put this on the scale of your good deeds.
We will answer your questions in the following points:
1- Regarding writing one verse or verses for a special purpose, like providing it as evidence or for teaching, it is Islamically permissible to write it in the common Arabic spelling. Shaykh Ibn ‘Uthaymeen said in ‘The Open Door’ meeting’: “If a person writes the verse as evidence, then we say: he may write it according to the known spelling rule, because if the people go back to the Mus-haf (the Arabic text of the Quran), they will find it written in Ottoman script.”
Nonetheless, we ask you to refer to Fatwa 93065 where we clarified that one should avoid writing the Quranic verses or Names of Allaah in such leaflets which are prepared for calling non-Muslims to Islam so that these verses or Names of Allaah would not be dishonored.
2- It is permissible to distribute these leaflets to people even to their homes, as the Prophet used to visit people in their homes in order to call them to Allaah. You are not obliged to obey your mother if she prevents you from doing so if she has no sound reason. For more benefit on the limits of obeying the parents, please refer to Fatwa 84942.
3- The backbone means the loin of a man. What is meant by the expression ‘the loin of Aadam’ in the Hadeeth, is what is called "the realm of atoms". Allaah The Almighty took all the people from the loin of Aadam like atoms and He took a covenant from them that He is their Lord, so they all confessed to it and then He put them back to the loin of Aadam.
Allaah Knows best.