A few thoughts....
"Lets say for an example, a person hires me to take pictures of her dog."
The initial contract is then between you and the person, unless you agree to give away your negs. or digifiles. As it isn't a corporate contract, and is one which is likely to be spoken and taken in good faith rather than argued out cogently on paper, you retain your photographers' rights.
What you will need to check is whether you have a model release form, or whether the model needs to give explicit consent for release. If the model (or the dog) is dead, the relatives can can act out of interest for the deceased. So 'further down the line....' is not always a safe bet.
If for instance, if the
www.beastiality.com website approach you for the images and you decide to sell the initial image to them without a model release form (from the dog and owner) then you are entitled to be sued american style. Similarly if the second sale of your image to an organisation enables it to be defaced, either by adding in a big turd where the owner has walked, or by adding an unkind caption, then similar unwarranted charges can be levied.
Another point of view which might be worth considering is that asking on an amateur forum is a recipe for disaster and offers conviction of viewpoint when a conviction in prison might be the realistic outcome. There are websites dedicated to the intricacies of the photographers' rights. Most photographers don't hire a personal lawyer to consult over the fine text of copyright, photographers rights and model rights etc so there is a degree of risk inherent in the work.
"I can do this with out permission from the owner of the dog right?"
---> Legal, but not necessarily ethical.
"I am not entitled to give them anything as far as profits off a picture they hired me to shoot right?"
No one would call this an entitlement. I presume you are thinking that if you subsequently earn from the images made in the first contract, then there is no duty of payment to the person who hired you. This is certainly how I work, although I endeavour to inform ex-clients that their work may be used by me in my capacity as a working photographer and that I will let them know if possible.
Good luck with it.
Good luck.