Well, literally "το κορίτσι" means "the girl" so contextually, as one may do in English by saying "my girl", the meaning may be "my girlfriend" or "my daughter".
However, there are also other Greek words to represent these meanings. Girlfriend -> η κοπέλα μου (my girl), η φιλενάδα μου (my girlfriend, although not too common in everyday speech), το ταίρι μου(my partner) Daughter ->η κόρη μου (my daughter), το παιδί μου(my child, contextually meaning the daughter)
Hope this helps.
There is no difference and both are used equally in modern Greek. There are some verbs in Greek called "συνηρημένα ρήματα" that have two ways of telling them. Verbs that end in -αω (such as αγαπάω) fall in this category. Εγώ αγαπάω/αγαπώ, εσυ αγαπάς, αυτός αγαπάει/αγαπά, εμείς αγαπάμε/αγαπούμε, εσείς αγαπάτε, αυτοί αγαπάν(ε)/αγαπούν(ε)
Yes αγάπη is the love for a friend, family member etc. Έρωτας is the erotic love and sexual passion you feel for somebody. Καψούρα is the burning desire for someone. Φιλία is the love for the friend (friendship actually) and στοργή the affectionate love you feel for someone, especially parents to their children. There is also φιλαυτία (if you want to add this to the list) that is the love for oneself, in a too much/ negative way.