The Question and Answer section for The Taming of the Shrew is a great resource to ask questions, find answers, and discuss the novel.
I think that Petruchio wants to show Kate that he is a man in charge and can command people. He wants Kate to be obedient to him.
Lucentio and Tranio tell Biondello that they're in disguise because Lucentio killed a man and needs to flee the city. They are disguised to fool any witnesses that saw the murder to throw them off.
The more I suffer, the more spiteful he becomes. Did he marry me just to starve me?