1. Home/Family|Home Décor are the categories used in the datafeed for all 791 products in the feed. So yes, the merchant did a poor job in categorizing their products.
3. Use only the merchant filter in your search to display all products at once. You need to create at least one category so you can pull the products in.
3a. Yes, the urls will be rewritten if you change the categories. Note however that the original category page(s) may already have been indexed when you do so! That's why it's better imo to create the proper store category structure now.