Hi Max,
Here are some suggestions for your 3 questions:
1. Search for some key words in an email in All Mail Items. This is to check if the emails are moved to any of your sub-folders by some local rules. If no, please check if you can see “Online with Microsoft Exchange” at the bottom of Outlook page.
2. The "Company Administrator" name is the name tied to the site collection administrator account that is being pulled from the Administration Portal. Therefore, I suggest you not delete it.
3. Please refer to Create a New Room Mailbox to achieve your goal.
Thanks,
Allen Zhang