While at the hideout, Desmond seems to insinuate that he's never had any experience as an assassin. However, in Assassin's Creed, Desmond retorts Dr. Vidic's statement that "he's an assassin", saying that he "doesn't do that, not anymore." Desmond also lived amongst assassins during his early years, it's only safe to assume that Desmond has some experience as an assassin, unless, as the story states, he ran away from the commune too soon to learn anything valuable.
Leonardo Da Vinci would have never been called "Mr. Da Vinci"; that would have been the equivalent of someone calling a person "Mr. From New York". He would have been properly addressed as "Leonardo." However, in the first Assassin's Creed Lucy explains to Desmond that the Animus translates the spoken language in the past to current day terms and grammar so that he would better understand the scenes.
(In Italian, "da" means "from", as in "from" Vinci; e.g., Leonardo da Vinci. "De" means "of" as in Lorenzo de Medici (of the family of the Medici). "Di" means "of" as in tintura di iodio.)
In the first Assassin's Creed Lucy is supposedly missing her ring finger. In Assassin's Creed 2 Lucy has all ten fingers. However, it is unlikely Lucy actually has a missing finger as the practice was eventually phased out of the Assassin rituals. More likely, Lucy simply bent her finger to symbolize to Desmond that she is on his side.