These might not fit all your criteria but they are all completed. Some of my favorites are:
Tsubasa, those with wings - Girl theif in a dystopian future teams up with an ex-soldier to try to exist and sovle the mystery of his past.
Kodocha (Kodomo no Omocha) - Young girl works as a child star and deals with school and friends.
Hana Yori Dango (Boys over Flowers) - A poor girl goes to the most expensive school in the country and has a run in with the F4, four of the richest boys in the school and the country!
Hana Kimi - American girl falls in love with a boy long jumper she sees on TV. Moves to Japan to go to school with him, only to learn he goes to an all boys school. She dresses as a boy to get in and becomes is roommate.
KareKano (His and Her Circumstances - A smart girl with a dual personality tries to out do and finally falls for the smartest guy is school.
Inuyasha - Girl falls down a well to the feudal era. She breaks a powerful jewel and must put it back together with a group of friends
Ouran High School Host Club - A girl breaks a priceless vase and must now maquerade as a boy to make up the cost.
Nana - Two girls named Nana meet and decide to live together. Shows their struggles in life.
Full Moon wo Sagashite - Girl dreams of becoming a singer, but can't because of the tumor in her throat. With the help of two Shinigamis, she is able to sing again.
Fruits Basket - classic, orphaned girl moves in with a group of guy that have a secret
Kaichou wa Maid-sama - super strict female student body president moonlights as a maid at a maid cafe
Special A - Tough, sweet girl consistently competes with the top guy in the school and interacts with the top students as well
Tokyo Crazy Paradise - Girl pretends to be a boy in a dystopian Tokyo. She becomes a body guard for the heir of a yakuza clan
Love Monster - A girl doesn't get into the school of her choice but ends up going to a school for monsters. She isn't what she seems.
Alice 19th - Alice Seno seems like a normal girl in high school. She is a bit shy, she has got a crush on a boy named Kyou, and she has got an older sister who is more popular than she is...pretty normal stuff, until Alice has an encounter with a mysterious and magical rabbit girl that changes the course of her life, and Alice is introduced to the sublime power of the Lotis Words.
Cat Street - A former child star is listless and directionless when a kindly principle introduces her to a school that can lift her out of her despondency.
Wish - A man frees a tiny angel and the angel promises to grant him a wish but he refuses. The stuborn angel tries to change his mind.
Chobits - A simple guy finds a Persi-com, human looking computers, and they interact.
Fushigi Yuugi - Two high school girl become trapped in a storybook and become priestess's of opposite factions.
Honey and Clover - Follows a group of students at an art college.
Angel Sanctuary - A brother and sister are the reincarnations of a set of ancient spirits.