Hummus is certainly Middle Eastern, and has always been popular in Syria, Jordan and Egypt. While it's certainly a staple in Israel, it didn't necessarily originate there. Like many Middle-East foods, it's virtually impossible to trace its origin as the cultures are ancient, movement and migration has always been fluid, and the ingredients are native to the entire region.
Many "Israeli" foods are borrowed from the surrounding Middle Eastern cultures as Jews migrated from these areas - Morocco, Iraq, Iran, Libya, Egypt, etc. -- and brought with them rich culinary traditions. "Jewish" food is generally thought of as the Ashkenazi variety from Central and Northern Europe: foods like kugel, kreplach, cholent, k'naidel, and borsht. They often have Yiddish names, as the people spoke Yiddish in addition to the various regional languages.
As is clear from the above, through the centuries Jews have been spread across the globe, and been part of various cultures, regional influences and have experienced culinary exposure. It's thus difficult to nail down any single uniquely "Jewish" food.
Answer:
It is believed that Hummus is mentioned in Bible in Ruth 2-14 when Boaz tells Ruth to come and dip her food into the sauce. The word used in The Bible is Humutz (Hummus) and similar to the modern Hebrew word Himtza. It appears from Biblical passages Hummus is mentioned as a Jewish food long before mentioned as an Arab food.
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.
Lamb can be kosher for Passover. However, Ashkenazi Jews traditionally do not eat lamb during Passover.
Yes, there are no restrictions on fruit for Passover.