Is there a way to turn off or hide the large unified mailbox?
Not exactly, no, as far as I'm aware. But for your purposes, sorta. Just click on the account you want to view the messages from in the Mailboxes list (you may have to click the little disclosure triangle next to "Inbox" to see it) and you'll only see messages from the inbox of that account.
Is there a way to stop showing mail from one of the accounts in the unified Inbox?
No, not that I'm aware of. That's what I'd like to do; I have several accounts. One of them is my daughter's. I want to be able to see her messages, but not have them appear in my unified inbox.
Is there a way to name the two smaller mailbox located below the unified mailbox as they both just read Gmail?
Yes. Go to the Mail menu and choose Preferences, then click Accounts. There you can rename the accounts without changing anything important.