•How many people (this includes programmers, designers, project managers, etc.) does this scope of a project usually involve?
•What would be the ballpark cost for developing this kind of app?
Assuming that you are the customer for this App these are questions better asked by sending out a request for proposals to several vendors who can produce this kind of app. Your best bet is to do a proper job of documenting your requirements and pre-screening a number of different vendors to identify a short list of who to send the RFP to.
If you are asking the question from the perspective of someone who is bidding on an RFP to produce this kind of app and are trying to get a competitive advantage, my advice is to not care about what other companies do or charge, build your response around your business case and not the business case of others.
•If the cost is above 10-15k, what would be the correct amount of work to give the client, if the client only can afford a very small portion of the original cost?
If the client can't afford the whole package break it down into pieces that are self-contained enough to be independent of the whole. Figure out what you will charge for each line item. Whichever of those line items the client chooses to purchase is what they can afford and is the correct amount for you to deliver.
•This is hypothetical, but let's say all of what was explained was near completed yet the client hasn't provided an IP agreement or any sort of contract, yet demands the work be turned over within the next week. Payment has been made but at a "very small portion" of the original cost, does the client have grounds to take all the work?
The first thing to do is to document in your "lessons learned" file to NEVER start work without a contract in place. Then communicate this lesson to the rest of your company so that they don't make the same mistake. It is always best for everyone to have a contract in place, in particular spelling out terms of payments (what each service costs, when payment is due, etc) and deliverables (clear description of each, what acceptance criteria are, etc).
If $$ has exchanged hands you may have a legal obligation to deliver something, depending on what jurisdiction you are in as in some a non-written contract could be as binding as a written one. You will have to negotiate with them to figure out what is fair based on what they paid. Depending on how those go you will need to make a business decision on the business value of fighting for the rest of the $$.