Imam Husayn (A.S) is a martyr – shahid. If Allah tells us in the Qur’an that a shahid is alive and not dead, why do we cry, beat ourselves and draw blood? It seems to indicate that, not only do we believe him to be dead but that he was helpless and unwilling to sacrifice his life?

Your question is based on the false premise that we weep for Imam Husayn (A.S) as we might over the loss of a relative.
The fact is that we share our glad and our sad moments with the Imams of the holy Prophet’s progeny. This is illustrated by the prayer of the 12th Imam (A.S),
O Allah, grant mercy to the followers who share their glad and sad moments with us.
Thawab al-Aamal by Shaykh Sadouq
One of the ways in which we strengthen our relationship with the chosen progeny of the
Prophet (PBUH) is by gathering together to joyfully celebrate their birthdays, in addition to gathering together in sorrow and sadness to remember their martyrdom.
Imam Husayn’s soul is conscious of what we do in his memory.
The souls of all human beings are immortal. Souls purified by martyrdom achieve spiritual levels which free them from any material framework. This affords them the awareness that is the essence of life.