re: an entry is listed under one of more categories
yes, an unlimited number of categories can be assigned to an entry.
re: Some listings will be loaded by us, others will be added by the business owner.
re: If an entry is added by us, we’d like the owner to be able to claim the listing
Sorry, no, there is no builtin in way or extension to claim a listing.
re: We’d like to feature profiles that have paid a fee and allow multiple images and longer profiles for these entries.
Sorry, but Connections does not have a way to pay for listing or make tiered listings.
Hope this answers your questions thoroughly.