The "gull", of course, is the trick played on Beatrice by her cousin Hero and her waiting gentlewoman, Ursula, inthe "pleached bower", (an arbor) (III.i.7) that misleads Beatrice into thinking Benedick loves her "so entirely" (38). This is a simple theatrical trick, in which the butt of the ruse is tricked into "overhearing" a conversation between two conspirators. This conversation leads the hearer to believe something that is not true (because the conversation is planned beforehand and staged by the conspirators) and, subsequently, to act on this wrong information. It is the basis of many of the plots of comedies and farces.
Beatrice wants to, deep down inside, believe that Benedick loves her, because she truly loves him. She remembers Benedick "of old" (I.i.123-4) and has, we are to believe, loved him, secretly, for a long time. This is also borne out by the fact that Beatrice turns down the marriage proposal of the Duke (in Act II Scene 1) -- a marriage alliance which would be envied by everyone, the Duke being the highest-ranking person in the whole company. Beatrice, if she were not in love with Benedick, would at least consider the Duke's proposal; but she turns him down flat. So, this revealing fact about Beatrice's character gives the audience a clue as to why she'd fall for the transparent "gull" of Ursula and Hero: everything in Beatrice has hoped for a revelation of this sort, and, psychologically, humans are very well-disposed to believe things that they have long wished to be true.
As far a surrendering her faults, Beatrice is nothing if not intelligent. She has always owned her faults, and been rather proud of her sharp tongue, quick wit, and brilliant (if sometimes too caustic) conversation. In order to win her heart's desire, she is only too willing to amend her fault -- who would not be, when faced with the possibility of getting the one thing in the world they want the most? ("Contempt, farewell! and maiden pride, adieu!/No glory lives behind the back of such./And, Benedick, love on; I will requite thee," III.i. 111-113).
As to Beatrice's true character, it is said many times in the play that she is "proud". Like many of the women in Shakespeare's comedies, she is proud, witty, and strong-willed - and very unwilling to be shown up by a man. This kind of character was a staple of drama at the time, and part of the elaborate game of repartee that makes up most of the love stories in the comedies of this era. Men are, in Beatrice's mind "constant never" (II.i.63), so her heart must be protected at all times. She feels that she has been "played" a bit by Benedick in the past, so she is wary of letting him know her feelings. In this, Beatrice shows her independence in addition to her pride. She may love deeply, but she doesn't want to be humiliated by it, in case the one she loves does not requite her. But, now that Beatrice "knows" that he does, she is free to love him and to stifle her pride somewhat.