In the Irish legend, did Aoife ultimately win or lose the duel with Cu Chulainn? Is it true that Cu Chualainn fell in love with Aoife?
I basically answer this question from the sources in another question here so I'll modify it for your question.
Aoife is a warrior woman who Cú's teacher Scáthach seems to have an ongoing feud with:
At that time also Scathach had a feud against other tribes, over whom was the princess Aife.
... she was afraid of Aiffe, because she was the hardest woman-warrior in the world. [Tochmarc Emire, Rawlinson B. 512, Trans. Kuno Meyer]
So naturally, he helps out Scáthach when Aoife comes to fight and goes against her in combat, though he won through trickery rather than combat when she gets the upper hand:
Then they fought upon the path, Cuchulind and Aiffe. Then she broke Cuchulind's weapon so that his sword was no longer than its hilt. Then Cuchulind said : « Woe is me ! » said he, « Aiffe's charioteer and her two chariot-horses have fallen dovvn the glen, and all have perished. » At that Aiffe looked up. At that Cuchulind approached her, seized her under her breast, threw her across (his shoulder) like a burden, and went to his own host. [Do Fogluim Chonculainn Annso Sios, Egerton 106, Trans. Whitley Stokes]
As for the loving each other bit, no way. Cú trades her life in return for her bearing him a son, medieval sources tend to include a lot of rape which gets scrubbed out in modern retellings.
... « Life for life ! » she said. « My three wishes to me ! » said he. « Thou shalt have them. » « These are my three wishes : thou to give hostages to Scathach without ever again opposing her, to be with me this night before thy own dun, and to bear me a son. »
It is granted thus and was all done. Then she said she was pregnant. She also said that it was a son she would bear, and that the boy would come to Erin that day seven year. And he left a name for him. [Tochmarc Emire, Rawlinson B. 512, Trans. Kuno Meyer]