Yes. Rav Moshe Feinstein even held that they are NOT "kitneyot" (a category which Ashkenasi Jews specifically do not eat on Passover) which means that all Jews may, theoretically, eat them on Passover.
However, since some Jews have the custom NOT to eat peanuts, it is unusual to find peanuts and peanut products with a kosher certification for Passover. Thus, although technically, eating them would be fine for most of us, practically speaking to actually do so while keeping kosher for Passover is difficult.
If you are considering spreading peanut butter on your matzoh, please remember that most commercial brands of peanut butter, even those with kosher supervision, contain corn syrup sweetners or other ingredients that are kitneyot.
It would have to be certified kosher for Passover.
The label has to say "Kosher for Passover"
Yes, only if they are specifically certified kosher for passover. You can find kosher for Passover olives at any Jewish supermarket
It can be, it would require a valid Passover hechsher.
Yes, it can be since it is made from wine, which can be Kosher for passover if created under appropriate supervision. The bottle would require a kosher for Passover hechsher.
If you were able to find corn vinegar that is certified kosher for Passover, it would be kitniyot which means that the majority of Ashkenazi Jews would not use it.
There are kosher food companies that are now making gluten-free products Kosher for Passover and all-year-round. However, something being gluten-free has no bearing on whether or not it is Kosher or Kosher for Passover.
Arugula is an edible leaf and is kosher year round including Passover.
Yes, they must be certified kosher for Passover though.
Some kosher companies make cereals that are Kosher for Passover.
Yes, there are no restrictions on fruit for Passover.
Lamb can be kosher for Passover. However, Ashkenazi Jews traditionally do not eat lamb during Passover.