Clarification. Asa and Ibuprofen cause thinning of the blood and not clotting problems.
Several factors. Lots of things can well beyond the scope of a 400 character site.
Help with diagnosis. Note: in general oncologists are also hematologist. They will help with finding an underlying cause for blood clots. Yes, cancer can be one cause but also lupus, antiphospholipid syndrome or certain risk factors like smoking, tamoxifen, elevated homocystine levels or acquired or genetic predesposition to clotting. Rheumatologist also help in these situations.
Heme/onc doctor. Many doctors who specialize in chemotherapy (oncology) also treat blood diseases (hematology) that are not related to cancerous conditions (most blood clots). Even so, cancer cells have abnormal dna that can lead to other systemic effects in the body as well as hypercoaguable states (blood clots).
Correlations. There is a distinct increase in risk of a stroke in those women who have a history of migraine with aura (classical migraine) who take birth control pills and also smoke. Why spend huge sums of money, get exposed to lung cancer or copd, or early heart disease, or as above, a stroke, and destroy your life? If you or close one smokes, get it stopped.
Multifactorial. Combination of factors. Add obesity and sedentary lifestyles in north america and you are at higher risk. It is common for european and increasingly asian females to smoke and be on the pill but they generally walk more and are skinnier and it does not appear to be a problem. Blood clots and strokes are bad when they happen - reducing risk factors is best.
Not necessarily. The bleeding can be caused from a number of different things, sometimes a bleeding disorder--however that is rare. Your doctor would need to examine you to see what is causing the bleeding and exactly where it is originating.
Yes. Consider an female condom, vasectomy for the male or using a condom, tubal ligation, iud.
Yes. In general, birth control that contains estrogen (the typical combination birth control pills, the patch, the ring) increases the risks of blood clots, and thus women who have a hereditary condition that increases the risk of clotting (factor v leiden & others) shouldn't take estrogen bc, but other options don't have estrogen: the shot, the implant, iuds, progestin only pills, and are ok.