I had a TSH level of 4.01 & have unexplained horseness and dizziness. With such a low score, could this be possible?

It's doubtful that your TSH has a direct effect on your voice & balance since it's the signal from your pituitary to your thyroid.  In fact, your 4.01 is not considered a low score as normal reference range is typically from 0.4 to 4.5.  So the most common way to interpet your "score" is that it's normal.  I would posit that if you're used to a TSH of 1, then a TSH of 4.01, while normal for most, isn't normal for you.  Think about a candidate for valedictorian who's used to getting straight A's.  S/he is going to be disappointed with that first B grade even tho most students would be ecstactic to receive a B.  Same thing w/TSH.

So let's take it a step further.  What if your TSH were actually increasing because your thyroid wasn't working properly and was actually developing a goiter?  That same goiter might irritate the nerves to your vocal cords leading to hoarsenes.  

Or perhaps you thyroid is overactive and causing your heart to race up & down w/o any control, eg atrial fibrillation with rapid ventricular response.  If your heart rate isn't right for you, that might lead to dizziness from poor cerebral perfusion.  That's just fancy speak for saying that you're dizzy because you're not getting enough blood to your brain because your atrial fibrillation is not efficient.

Best to go back to your doc, the one who ordered your TSH, and ask his/her opinion over your concerns.  Good luck!