Hello, nice project you have there, I surely can help you with it. And since you're a charity I offer to do it for a fraction of my usual price. I'm a pet lover too and participated in a re-homing group myself when I was back in my home town :)
I'll use Twitter Bootstrap front-end framework, which is mobile friendly, so it will look good in tablets, however, keep in mind this will be a web app, not a specific mobile app. But don't worry, it will work flawlessly.
One question, you said:
"""
There should be a dropdown box with options for "Available for rehoming, rehomed, reserved". The choice on this dropdown should determine which tab/page the pet ends up on
"""
I guess that's an option only for site administrators, right? That it shouldn't be available for not logged in users, correct?
Also, you didn't specified if and the means for users viewing the page to adopt a pet. I'm thinking we can change the specification to show pet information on click and move it to an on hover gesture, while keeping the click action to lead user to a view which let's them adopt the pet and insert contact information. But that's just an idea, I don't know how does your charity work, maybe you prefer to be called on the phone for that part.
Finally, I cannot program "etc", if you want me to show more information on the data view of the pet, the please specify it, I need it to properly devise database and view and, of course, to properly estimate time and cost of the project.
Thank you.