As legumes, peanuts belong to a subcategory called 'kitniyot'. As such, they can be kosher for Passover, however, Ashkenazi Jews will not eat them.
Yes, they must be certified kosher for Passover though.
It depends on the brand. However, even brands of real mustard that are certified kosher for Passover are still kitniyot.
It would have to be certified kosher for Passover.
Yes, Almonds are Kosher. Raw nuts do not require Kosher certification. Nuts that are oil roasted, dry roasted or seasoned, both in or out of the shell, require Kosher certification. An exception would be peanuts roasted in their shells. They are acceptable without Kosher certification. Peanuts aren't really nuts, they're legumes.
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, if it's labeled as kosher and they're not allergic to peanuts.
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.