Asymmetric nose normal? (
Yes, your nose is asymmetrical which is very normal. Yes, your nose can be refined and it will change your look very slightly. I do want you, however, to look at your cheeks. They are very flat. Adding angularity and balance to the cheek area will take away the focus on your nose. You stated you didn't want to alter anything about your nose...so, you really need to consider whether or not you really want surgery on your nose.
Your nose i asymmetric, slightly. Is it grossly asymmetric? No. Is it normal for you? Yes. Is it normal in general? Generally yes.
Your nose is asymmetric due to the asymmetry of your face (this is quite common, by the way). Your upper jaw is moved slightly to the right and, as the nose is attached to the upper jaw (the maxilla), your nose developed such that it is also deviated to the right. With this, the nostrils are made asymmetric. The best way to fix this is to move the jaw to the midline, but this is extensive surgery. Rhinoplasty surgery alone can make it look much better, however.