Yeah, in the first cutscene of DmC2, Lucia already knew who Dante is (she called him "son of Sparda"), but I take it that that was the first time they've met in person because Lucia really thought Dante was going to shoot her and defended herself right away.
I think they've had contact before hand, and agreed to meet at the museum where Lucia points out their (her's and Matier's) location at Dummary Island.
If I remember correctly, the only time anyone said Lucia's name is at the beginning cutscene of Dante's last Mission. The player learned about Lucia's name before hand by reading the back side of the game's CD cover.
Also, the mystery girl was already revealed a couple of days ago in the DmC4 Info HQ thread, and I was not Lucia (it's a new comer named Gloria)
"Alot," isn't a word, don't merge the words "a" and "lot" together.