IOS – Why do some contacts on the favorites screen on Phone.app show initials and others an icon

Currently, no.

That is the way it works in iOS 6 (if you ask me, it's a better way).

But...

This month, there should be a massive redesign with iOS 7 - maybe this design will be improved/changed.

I suffered from the email-address-instead-of-name problem after replacing my phone's local contact info with a sync copy from my gmail account. this seemed to work for me (ios7):

1. click on the contact name in the 'find my friends' window
2. click the 'contact' button at top left
3. click 'add to existing contact'
4. select the relevant person in your contacts list; this will seem to add a duplicate email address, and you might want to clean that up later

restart Find My Friends... is now showing names (for me at least)... good luck.