Yes it is common for migraine symptoms to be cold hands and feel. I can often feel cold all over- like an ice cube! It must be from the blood vessel constriction.
Getting rid of migraines can be done by serotonin like replacement drugs like imitrix or by enough time going by that your body will naturally find the proper serotonin level for your body. Relaxing...
It can last for any amount of time. It generally can last from a hour or two, from the first symptoms a migraine (these can be very different for different people, auras, sounds sensitivity, etc. )...
Migraines are sometimes caused by cold food products. Have you ever had a headache eating slush? In some people this can cause a migraine.
Migraine headaches are induced differently for each person...