IPhone – Is it possible to sync messages from Mac to iPhone

I think the source of your confusion is the difference between traditional text messages (SMS) and iMessages (Apple's own protocol).

Messages on the Mac doesn't support SMS. Any message you send or receive on your iPhone as SMS won't show up in Messages on the Mac.

If you send a message as an iMessage, though, then it'll show up on the Mac. Now here's where it gets confusing: even though iMessages are "data" and not "text," they can be sent between phone numbers. If you've got iPhones at both ends, a "text" sent from one to the other will be sent as an iMessage by default. These messages will then appear in Messages on the Mac.

No. Only iMessages sent from the time you set up Messages on your Mac can be received.

Note that you are not actually syncing between Messages on iPhone and Messages on Mac - you are adding a new device to receive a copy of each message sent. When you register a device for iMessage, a set of keys (a public key and a private key) are created. The public key is sent to Apple, while the private key remains private and known only to your device. When someone wants to send you a message, their app contacts Apple and requests your public key(s). If you have four devices registered, they will receive four public keys.

At that point, their device encrypts the message once for each public key. Only your private key for a particular device can open the encrypted messages intended for that device. So if (by some technical problem) the copy of the message intended for your iPhone was sent to your Mac by mistake, it won't be able to be opened, even though it's for you and was encrypted with your public key.

Hopefully this makes it clear why messages sent prior to registering your MacBook with iMessage cannot be retrieved - simply put, they don't exist in a form that can be read by your MacBook. Even if Apple kept a copy of the messages at the time they were sent (which they for up to seven days, or until the message is received by the intended device, whichever comes first), only your iPhone can decrypt them.

This page has a good overview of iMessage encryption (which goes into more detail on what I've outlined above).