First of all, keep in mind that I am unable to diagnose you because I am unable to examine you, this forum is for educational purposes.
I agree that from the story you provided it sounds like your father had a small stroke in a deep area of the brain that damaged some of the 'wires' that control the right side of the body. The stroke was almost certainly due to his hypertension. Hypertension is the leading cause of stroke, and causes narrowing of small ateries that feed the deep structures in the brain (such as thalamus, internal capsule, basal ganglia, etc.) a process called lipohylanosis (even with medicine). High homocysteine is thought to contribute to the risk of vascular disease, but the exact contribution is less clear. I agree with starting the B vitamins and the aspirin. However, I would suggest some more tests. First, you need to have an idea about the health of your cerebral vessels. This can be obtained by MRA (magnetic resonance angiography), CTA (Computed tomography angiography) or by carotid duplex and transcranial ultrasounds. If there is a stenosis, then the risk of recurrent stroke will be higher. Second I would recommend a echocardiogram of your heart, to assess your ejection fraction, the health of your valves (it sound like there may be a problem from your post), and to evaluate for a PFO (small hole in the middle of the heart). I would also suggest that you wear a heart monitor (HOLTER) for at least 48 hours to make sure your heart is not beating funny (atrial fibrilation), because this can lead to strokes and requires a different medication (coumadin). You should also have your cholesterol checked and take a statin if needed. Ulcers in the mouth are not a problem with aspirin. Aspirin is only a problem if you ave ulcers in your stomach or GI track. Driving is very often possible after a small stroke, but I would suggest that you see an occupational therapist that specializes in driving assessment to make sure everything is safe. I would focus on finishing your workup with the tests I described, diet and exercise, address your cholesterol as needed, take daily aspirin and control your blood pressure as well as possible.
I hope this has been helpful.