How do I keep my cuticles neat?

I have a habit of picking at my cuticles with my hands because they keep lifting up.

I do. ㅠㅠ As a result, my hands have gotten very dirty.

Would it be good to remove the cuticles once a week?

Are there any nourishing products for cuticles? My thumb, in particular, tends to peel easily.

I keep picking at it... T_T It's hard to manage.