Coughing normally will not cause bleeding in ears. It can however cause problem to an extent if your eustachean tubes (connecting nose to middle ear) were blocked due to cold.
From you history, there seems to be some injury or boil to your ear canal
. Your hearing seems to be ok since you have not mentioned about any loss. Please get your ear examined.
There is no need to suffer from a cough bout for 1 hour! Whenever your asthmatic airways trigger a cough bout, it is a good idea to take inhaled salbutamol
1-2 puffs and that should abort the cough bout.
Bad prolonged cough
bout can cause neck/ abdominal pain
and can also cause fainting