Deep lines under my eyes only when I smile, would Botox have any effect? (photos)

Since these lines are due to dynamic facial expression, neuromodulator, such as Botox and Dysport, should work very well to mitigate them. The amount of neuromodulator should be conservative, as over-treatment in this area with neuromodulator will cause lower eyelid retraction and thus dry eye. 

Wrinkle Blockers under the eyes (Xeomin, Botox, Dysport) can work fantastic for wrinkles like you've got.  If you smile very expressively they will still show to a certain degree, but I would expect significant improvement with good placement.  Medical grade skincare like A.G.E. from SkinCeuticals also helps.  Starting young helps your skin keep from getting deeply wrinkled and creased, so you are planning well.  Best regards.