It's one of the proposed etymologies, but it's more likely that pasta comes from the Greek word for a thick porridge that came to refer to dough, and is at the root of the English words paste and pastry, as well as the French loans pâté and pâtisserie. Pasto instead comes from the Latin pastus, the past participle of pascor, "I eat".
Yes, that would be possessive. This sentence is not showing that it is "yours." Also, the sentence you put isn't technically bad English. It is not used often, but it is sometimes used as a suggestion. For example: My kid says, "I'm bored!" I respond. "Why not finish your puzzle?"
Zero knowledge on the grammar/subject here... but there were a bunch of questions about the progressive before and how the 'ing' versions of English phraseology are not always applicable (at least with the Italian words supplied). I am not sure I followed what I wrote there. I blame il vino. There, that one is free for the listeners :)