Question One: "Why did King Charles impose tax and request new funds through extraparliamentary measures? What limits does the Petition attempt to place on royal taxation?"
There was one main reason for King Charles taxing the people of England without Parliament's consent. This was that he and the Duke of Buckingham George Villiers had insisted on fighting a war with both France and Spain. He needed money desperately so that he could fight this war and because of this The Petition of Right came to be in 1628. It was because of his foreign policy that he needed money. It was because of Parliament that he was not able to tax the people in the first place. Even though Parliament forbade it, he still taxed the people without their consent so that he could fund his wars. Initially Charles I had agreed on signing this document but he later changed his mind after realizing it limited his power.
To try to limit royal taxation they say in the Petition Of Right that "no man hereafter be compelled to make or yield any gift, loan, benevolence, tax or such like charge without common consent by Act of Parliament, and that none be called to make answer or take such oath or to give attendance or be confined or otherwise molested or disquieted concerning the same or for refusal thereof." They were basically saying that no person should have to give money to the king without the king having Parliamentary consent. The limit they were trying to place upon the king was that he would have to ask Parliament before he could tax the people for things such as funding a war.
Question Two: "How did the Petition criticize arbitrary arrest?"
There were a couple main reasons the Petition criticized arbitrary arrest...