Can you also say רוצי לחוף?
yes, you can
Yes this is feminine. Masculine would be רוץ
"run towards the beach" is wrong? Why?
Is the verb not conjugated according to gender? Is the י not signaling that it is to a woman?
Yes it does.
רוצי for singular female
רוץ for singular masculine
I thought the word is pronounced "al" not "el"?
You're probably confusing with another word.
אל (el) = to
על (al) = on
אל (al) = Negation of imperative