Hey there,
Bit by Bit Development is a small mobile development shop based in Canada. We've been working with mobile applications since the very first iPhone was released. We're confident that we have the skills and experience to help you build out the application you described in a relatively short period of time (two weeks).
With that said, I read through the description, and it seems like you have most of the heavy lifting done already. Design and APIs are (in our experience) often the most time consuming parts of a project. I'd love to take a look at some of the designs you have, just to confirm and validate the general idea and picture I have in my head from what you've described. From there, we can confirm the price and timeframe, though I don't really expect them to change.
One project we've worked on that has most of the core concepts you're looking for is Say the Same Thing. It's a simple game we worked on in the past, but it uses custom lists, Facebook login, sharing, all sorts of different setting toggles, and a backend REST API. It doesn't touch on some of the graphical elements you're looking for like carousels, but carousels as you described are a pretty native iOS widget, so they can be added and customized pretty easily.
I'd love to follow up and have a quick conversation about this project, and feel free to ask any questions - we'll get back to you ASAP.
Looking forward to hearing back from you.
Thanks,
Justin S
Bit by Bit Development