Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
As for performing prayers and teaching the Qur'an, it is a condition for a Muslim to be with complete purity before doing any of these two actions. This is the opinion of the majority of Muslim scholars.
As for blood of wounds, it does no harm as long as it is in small quantity that does not break Wudu. However, you have to wash blood, however little, before touching the Qur'an since it is impure.
Finally, there is objection to touch and listen to magnetic tapes of the Qur'an.
Allah knows best.