Re: If a pet already gives a card, can I add another?
Some pets come with a card at baby, at ancient ect. those aren't skills/talents, they don't fill a slot. The game will look to see if your pet has a talent in its slots that has the same name as the one given by your jewel and if they match then it will turn one off, the text will turn red and the talent will be listed as not usable. The card that your pet has because it happens to be a siren shouldn't count.
If you are asking if a pet can give you multiple spell cards to cast, then yes, my penumbra drake currently gives me five different spell cards, and is only an adult, with no jewels socketed.