Sara,
I can tell you from experience (I was an EMT for 11 years), that emergency personnel are not going to look at your driver's license. If you're unable to speak for yourself during an emergency, they're only going to be worried about you breathing, and/or bleeding profusely. Facial paralysis wouldn't really change the way they treat you in an emergency, and chances are, if you're unable to speak for yourself, you're probably unconscious and your facial paralysis won't even be noticable.
Once you get to the hospital, they may look at your license if they need to know your name (or, sadly, to see if you're an organ donor, but if that's the case, I don't think they'll care about your face), but they are still required to run the same series of tests that would not be affected by your paralysis.
Ideally, emergency personnel should know if a person is on blood thinners though, because you wouldn't want them to keep poking her with needles and it would be helpful to know why she might be bleeding more than would be expected from what might be a small cut. But if she happens to be bleeding a lot, they're going to treat it the same way they would if she wasn't on blood thinners.
I carry a card in my wallet with all of my medical history and current medications and the name of my doctor, just in case. My husband also carries one for me, because I know if they asked him in an emergency what my medical history was, he'd probably forget. I don't expect that any emergency workers will pull it out of there, but my family knows it's there if they need it. It's also helpful when I need to tell it to a new doctor or something, as I can't always remember all of it either.
But, all of that being said, if it's something that makes you and your family feel more comfortable, then by all means - you should do it. It's certainly not going to hurt. Or, you could just try to avoid being in an emergency in the first place. That's a much safer option!
Lori