Apart from the base char modifiers, does your current location or equip (such as items granting special chars like Bizarre or Respectable) influence the opportunity cards you draw?
Opportunity cards are added to your deck based on whether you qualify for them with your current qualities; this includes the main qualities like Persuasive and Shadowy, but also story-related qualities like Admirer of Art, Touched by Fingerwork, and Scholar of the Correspondence. While you can freely affect your main qualities via equipment (and you will qualify for stories including the effects of equipment), you can't affect story qualities via equipment. You can also change your Bizarre, Dreaded, and Respectable via equipment, but there are no opportunity cards that are unlocked by those three qualities.
Owning certain items can make you qualify for particular cards. Equipping an item does not have any further effect. The earliest examples are pets: the Sulky Bat, the Deshrieked Mandrake, and the Reprehensible Lizard will each add a new card to your opportunity deck if you own one.
Most locations will alter the opportunity deck, too. Various places around Fallen London proper might add a few cards to the opportunity deck. The full list is on one of the wikis, but, for example, Watchmaker's Hill has the cards "A tavern dust-up", "Deal with a drunk old soldier", and "The Departed".
A handful of locations will completely replace your opportunity deck with entirely different cards, first discarding any face-up cards you have. The earliest example is the Pickpocket's Promenade in Spite, but there are many others: sailing on the Zee, Polythreme, each of the Places of Menace, etc. In those strange locations, none of the opportunity cards unlocked by your owned items will be in the opportunity deck.