Yes. Dog food is probably chametz on Passover and may not be owned by a Jew on Passover. Most dogs will happily eat Matzo and meat/chicken on Passover.
There are two options for Passover, you can cook food for your pets or you can purchase kosher for Passover pet food. Please see the related link for a list of approved pet foods.
rid their households of all leaveaned products
Science Diet dog food does not have chumetz (leaven) in it, but it does have kitnyot (legumes). Since the food is created in a factory that is not supervised for Passover, you should buy the food before Passover so even if there is 1/60 chumetz, you are safe.
They get rid of there food by regurgitating it.
Yes.
I'm sure they donate money and food.
Either the leaving of Egypt, or the slavery of Egypt.
get rid of their food supply
No processed food is kosher for Passover unless its label has a symbol of reliable Rabbinic supervision.
Food is cooked no differently than at any time of year. We just use some different ingredients (only food that is kosher for Passover) and those who are observant will use different sets of pots and pans, dishes, etc.
It was the Feast of the Passover, a traditional gathering with food. Jesus used the symbolism as He became Our Passover - a blood sacrifice that could and does forgive sin and the guilt of it.