Question: Which Outlook inbox should get new Gmail messages?
Messages received by an IMAP account should always show up
under that account's dedicated inbox.
Question: Without using a filter, is there a setting to direct Outlook to use the inbox I choose?
No, because it's not Outlook which stores messages in that inbox. As you're using an IMAP account, Outlook only gives you a direct view into the folders and messages that exist in Gmail – it doesn't actually collect mail from there like a POP client would.
If you used a filter to move messages from Gmail's inbox to a local Outlook inbox, 1) they would actually be removed from Gmail's inbox, affecting the website as well, 2) you'd be basically reinventing POP clumsily. If an independent local copy of all messages is what you want, it might be more appropriate to just use Gmail through POP.
Question: Is it normal for sent messages to appear in both Personal Sent and Gmail Sent folders?
Yes, the first copy in "Personal" is stored by Outlook before sending, while the second copy is stored by Gmail itself when the message is actually being sent.
This is specific to Gmail – as far as I know, most other mail servers do not automatically store sent messages this way, relying entirely on the client (e.g. Outlook) to do so. Which is why Outlook does it by default.
You can disable the former (in Outlook's account settings) but not the latter.