I suggested the aspirin because you said you did not have health insurance and thus did not want to go to the doctor. Aspirin is relatively inexpensive and easy to get, and it is something that has been recommended to me by a doctor.
Since menstrual blood has already clotted, I should think that taking aspirin might help things in that menstrual clots can be a source of additional cramping. In any case, you can always counteract any possible blood-thinning effect by consuming foods high in Vitamin K, such as romaine or leafy green lettuce, cabbage, broccoli, and kale.
I was first told to take aspirin in advance of my period by my gynecologist because I have always had severe cramps. She said that if you wait until the cramps have already started the prostaglandins have already been produced and so aspirin will have a lesser effect. She said to start 24 to 48 hours before I expected my period to start.
However, when I hit my forties and was in perimenopause the remaining estrogen in my system caused my fibroids to enlarge, causing even more severe cramps and heavy flow. In that case, menopause itself solved the problem as without estrogen the fibroids shrank. However, there are estrogen agonists you can take.
As to drinking cold water, by the time the water hits your stomach it is already warming up, and by the time it has anything to do with either the blood stream or the uterus it is at body temperature.