The devices Mac book pro / iPhone / Mac mini are signed in with same Apple ID and password on the same network , your wife iPhone is visible as find my iPhone is turned on Set up Find My on your iPhone, iPad, iPod touch, or Mac - Apple Support , the avatar image can be similar , if she had added the same image as in your iPhone ( it will show same image in appleid.apple.com ) , but the device name will be different .
She is changing first and last name of hers at appleid.apple.com ( clicking on edit and then changing and click on done to save an immediate mail appears in the signed Apple ID account in Mac that is in IMAP account ) .
Her Apple ID account credentials will be always different from yours including Apple ID and password , and security answers and questions , date of birth , rescue mail address and its password , if the account is set as secondary authentication .
While using air drop between devices , open air drop from spotlight , it will be prompt to connect blue tooth , click on drop down arrows , if it is selected for everyone so it will be visible in all devices , it depends upon you to choose the desired option .
Also find my Mac is enabled in Mac system preferences Set up Find My on Mac - Apple Support , so devices are located using location services Locate a device in Find My on Mac - Apple Support
To your question ( a ) she is first using air drop with the connecting devices , from her iPhone that's the reason her name is showing , regarding common avatar image it can be switched at iCloud.com Change your Apple ID profile picture on iCloud.com - Apple Support
On Mac Select your Apple ID picture on Mac - Apple Support
See a screen shot for drop .

For Mac system preferences , find my Mac .
