Does my crooked smile have anything to do with the way the tip of the nose sways to the right with my smile?

Asymmetry is very common in the face and does include the nose. When a strong upper lip muscle is present, this can deviate the tip of the nose to the stronger side. To make the nose more symmetrical, osteotomies are usually required of the nasal bones, a spreader graft is placed under the concave upper lateral cartilage and tip suturing techniques of the lower lateral cartilages are all employed to make the nose look more symmetrical. A columellar plasty is performed when the columella is crooked as well. Please see the link  and the video below to our rhinoplasty photo gallery for many examples similar to your own.

The commissures of the mouth are not affected by the nose or vice versa. The central lip can sometimes be affected by a muscle connected to the septum. Find a plastic surgeon with ELITE credentials who performs hundreds
of rhinoplasties and rhinoplasty revisions each year. Then look at the
plastic surgeon's website before and after photo galleries to get a
sense of who can deliver the results. Kenneth Hughes, MD Los Angeles, CA

No one is born with perfect symmetry. As you can see in your photograph, one side of your face is shorter than the other. You can also see that your nose deviates to the shorter side when you smile. So, to answer your question, your crooked tip is associated with your crooked smile. You can certainly undergo surgery to make your nose a bit straighter. But, there are facial muscle dynamics that can not be changed with surgery. If your nose is straight when you are not smiling, but crooked when you smile, your nose will continue to pull to the shorter side when you smile after surgery. So, would it be worth it to you to undergo surgery? Thank you for sharing your photo and questions. Best wishes.