I heard that it's bad to cut your cuticle...but whenever I don't my manicures always turn out looking sloppy?

Like the polish near my nail bed always turn out looking really bad. Sometimes i'll accidentally get polish on my cuticles. How can a prevent this from happning without cutting my cuticles and risking an infection? Also the skin around my nails always look scraggly and dry. Is it okay to cut that skin off?