Welcome to Bitcoin!! I'll try to answer some of your questions:
A bitcoin isn't exactly a nonce or hashed block (including the nonce) found by instense number crunching. Instead, think of a bitcoin as just an accounting record, like a number you might type into an excel spreadsheet to represent the amount of money that some account has.
Now, when when people transact bitcoins, they broadcast their transaction to every node on the network. Over time, a few of these transactions will build up. Here's where the computation comes in. All of the node computers will begin processing this "block" of transactions which have occurred since the last proof of work began, looking for a nonce that happens to generate the target when hashed. When a node successfully finds this nonce, there is essentially just a new accounting record that says that node computer gets a 50 bitcoin reward, which can be spent in future transactions.
One of the really weird things is that the number of transactions that need to be "blocked" doesn't actually affect how hard it is to find a nonce for that block. So, there's really no benefit to not including the transactions when searching for a nonce, like you might have thought when you said "...concentrate on the number crunching".
Now, the rewarded bitcoins must go to the node who found the nonce, because along with the transactions and the nonce, the block also includes the node operator's address. So, even though the proof-of-work can be publicly verified, nobody can "claim it as their own" because that nonce is only a solution for the block which includes the successful node's address.
I hope that helps clarify some things. Please continue to ask questions if it's not clear, this thread can be a great resource for other newcomers. Somebody else will have to clarify how bitcoins are divided, I'm not entirely clear on that myself.