My understanding is that "incontrare" means "to meet" as in "I hope I meet Fred on the street when I walk to work so I can give him back his hat." If you were saying that you wanted to meet someone whom you had never met before, as in "I want to meet your new girlfriend", then you would use "conoscere". Is this correct?
Do you mean, for example, "I want her to meet my father"? If so, you don't structure it this way in Italian. Rather, you say "I want that she meet my father" -- "Voglio che lei conosca mio padre" (or, if you mean "meet" as in "meet in the street while out walking", "Voglio che lei incontri mio padre.")
In using this construction, note that the second verb is conjugated in the subjunctive (conosca instead of conosce and incontri instead of incontra).
Not in this duo sentence however. It will be marked wrong. Duolingo has a policy that if they are using the polite form it will be capitalized, even though that is not the general rule in Italian. They do this so that the formal you does not confuse new learners. There is a section later in the Italian tree dedicated to the formal you.
The problem is that the Duolingo engine can't read your mind and tell why you missed it. The la ("her") is right there in front of you; Duolingo assumes that if you're told that you are wrong and given the correct answer, you can figure out for yourself where you messed up.