Coumadin (warfarin) is a very common anticoagulant drug, used if someone has had a stroke, heart attack, or deep vein thrombosis, to make clotting in the blood vessel system less likely (to avoid a clot in the heart, brain, etc.). While it is a lifesaver, it is also scary, just as you describe.
Perhaps you need to reframe it for yourself. Because she's on coumadin, now, your mother will be VERY closely monitored. While she's "getting used to it," she'll go in frequently to have her coagulation (clotting) tested--the MD will increase or decrease the coumadin until she becomes "steady" at the value he wants. After that, she'll go in less often, but will still be checked monthly, probably.
It's a life-saving drug, that MANY people are on (you'll be amazed, if she shares it with others, how many will respond that they are on it, too).
Here's an article for you. You might encourage your mother to wear a bracelet that says she's on coumadin, so that "first responders" will be alert for the possibility of hidden bleeding if she's in an accident (God forbid). Notice the dietary recommendations (avoid grapefruit; easy on the broccoli, etc.). http://www.webmd.com/drugs/drug-4069-Coumadin+Oral.aspx?drugid=4069&drugname=Cou
It's not that she cannot exercise (unless there is something else going on). She'll need to work with her physician regarding low impact exercise--perhaps swimming.
Whether she's on it forever depends on what she's taking it for. My husband has a history of stroke and heart attack; he's on it for the duration. Again depending on what she's taking it for, NOT taking it (and having the stroke, heart attack, etc.) could shorten or end her life; so taking it is more likely than not to extend it.
Besides, she's got you!

) Best wishes.