Jersey City Food Spots
I can only speak for downtown JC, and as a vegetarian, I can't help you with the first 2 requests.
I think the best grocery store is the ShopRite at Metro Plaza on Marin Blvd.
For the gourmet stuff, you might want to try Hoboken. Although for stuff like bread, cheese and meats, you can hit the farmer's market in Van Vorst Park on Saturdays; for the rest, I go to the small grocery story on Grove St. close to the PATH. The produce is cheap - enoki mushrooms for $1 - and they have a fair amount of higher end brands.
I don't live in Jersey City so can't speak for the local places. However, you are just a 10 minute PATH train ride from lower Manhattan NYC, home to many culinary treasures.
Take the PATH train to 14th street, and check out the Union Square Greenmarket, an outstanding outdoor farmers market for the best local produce and food items. Its open year round and offers everything from vegetables and fruits, to breads, eggs and dairy, meats and seafood, flowers and herbs, and other food-related items.
Also in the vicinity of 14th street, check out Chelsea Market (actually 15th and 9th) a series of wonderful gourmet food shops in a funky indoor setting, including bakeries, italian market, produce, fish, wines, and restaurant establishments.
A new version of Balducci's also opened recently on 14th street at 8th Avenue, which is a famous gourmet market, prepared foods, and wonderful butcher shop.
Citarellas is another wonderful gourmet market located on 9th Steet and 6th avenue, right next to another PATH train stop. They feature outstanding fish as well as gourmet products and prepared foods.
Between these 4 places, if you can't find it, its probably not in existence or in season.