Sync contacts between different account types
After doing some research it seems it's impossible to do this using stock Android software.

Sync can be done with Contacts Optimizer. This should transfer basic contact information between account types, however leaves out picture information (at least I've found it to copy low-res photos only).

So I've tried apps like Sync.ME and CoSy for pictures (at least to get the latest info from social media sites). The former was a miss - it doesn't actually store the downloaded FB information as part of the existing contacts. The latter seemed to work much better, and actually stored high-res photos as part of the contact information.