It depends on the brand. However, even brands of real mustard that are certified kosher for Passover are still kitniyot.
Many Jewish communities don't eat peanuts on Passover. For those who do, it needs Passover kosher certification.
No.
1) You should ask your own Rabbi, because there are different customs regarding this. 2) Generally speaking, Ashkenazim (Western/European Jewish heritage) do not use any form of mustard on Passover. 3) For Sephardim (Eastern heritage), mustard in any form should be purchased with proper labeling as being kosher for Passover. Otherwise, the anti-caking elements (stearates) may be from a non-kosher source. If the dry mustard lists no ingredient other than mustard seed, some are lenient and consider it kosher for year-round and for Passover without Rabbinic supervision, but others are stringent.
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.
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.