What version of iOS and OS X?
Is wifi and Bluetooth on (I don't think Bluetooth is required but for testing purposely lets just leave it on on both devices)?
Are you on the same wifi network?
Is continuity working at all? Meaning is there an icon popping up on your Mac to the left of the dock when using an Apple app? You can test this by playing in Safari on your phone while sitting in front of you Mac.
......
I'm not too sure about the contact issue but a couple of things.
If you have the contacts in the contact app on your Mac they should be syncing to the iPhone. If not then there is a syncing issue which could be part of your above problem we'll need to sort out.
Also if your contacts are synced with Google/gmail you should be able to goto settings > mail, contacts, calendar and sync them to your iPhone from there......I believe....