Though you are technically only allowed one question per submission, I have only edited out one of the three that you proposed, as the two above relate to the Duke and his point of view on the elopement of Desdemona and Othello. Please re-submit as individual questions any other questions you have about the Duke and Brabantio.
These quotes are from Act I, scene iii, and relate to Brabantio's claim that Othello has "stolen" his daughter Desdemona, making her elope with him.
The first quote you cite comes in this line of the Duke:
I think this tale would win my daughter too.
Take up this mangled matter at the best;
Men do their broken weapons rather use
Than their bare hands.
First, it must be noted that these lines are in direct response to one of the most famous speeches in all of Shakespeare -- Othello's recounting of how he, as a welcomed guest in Brabantio's house, wooed and won the heart of Desdemona with his tales of his adventures. This speech reveals the poetic, eloquent and noble nature of Othello, and casts Brabantio's harping and false accusations in a very poor light.
In the line cited above, the Duke remarks upon this, suggesting that the situation might be a "mess," but that Brabantio should cut his losses, since his accusations or "weapons" have been "broken" by Othello's speech. His line ends with a veiled caution that Brabantio probably does not want to tangle further with Othello in bare "hand to hand" combat.
Once Desdemona has confirmed her free choice of Othello, Brabantio reluctantly gives up ("I have done, my lord.") and the Duke has a few final thoughts about the matter for Brabantio:
To mourn a mischief that is past and gone,
Is the next way to draw more mischief on. . . .
The robb'd that smiles, steals something from the thief,
He robs himself that spends a bootless grief.
Clearly, though Brabantio says that he is "done" he seems headed towards harboring a grudge against Othello and his daughter. The Duke, in the lines above, cautions him to let bygones be bygones, because clinging to "mischief" from the past will surely bring more "mischief" or misery. He creates an image of robbery that calls upon the Christian notion of turning the other cheek, proposing that Brabantio, though he feels he has been "robbed" of his daughter, "smile" anyway and save himself from causing his own future misery by hanging on to a pointless ("bootless") grief.
For more on this scene, please follow the links below.