Is Titus' madness feigned or truthful? Or both? How do other characters respond to this madness?
I think his madness is feigned and truthful. Clearly, with all the bloodshed, he's gone a bit crazy, but at the same time, he's just doing what he can to save what's left of his family. For example, when he asks Aaron to cut off his hand, he does so to spare his two sons; but when he finds his two sons have been executed anyway, his reaction is laughter - so, here he's gone a bit mad but he may also be feigning. Aaron probably thinks Titus has lost his mind. His relatives believe he's mad when he asks them to shoot arrows into the sky asking the gods for vengeance. But these of course are part of his plan for revenge.
When Tamora disguises herself as Revenge, she may think Titus is mad, but he reveals to the audience that he knows exactly who she and her sons are, disguised as (according to Titus) Rape and Murder.
His slaying of daughter Lavinia seems cold-blooded and insane: here is an example where he is mad and calculating. Lavinia may want to be dead after what's happened to her and/or Titus may want to erase her shame, but Titus kills her primarily to let the others know that he knows she was raped. The scene where Titus reveals to Tamora that she's eating her sons' remains also reveals Titus' madness and calculation. Although much more bloody, Titus is similar to Hamlet in that both go to great lengths to reveal wrong-doing in the most dramatic way.