MacBook – Can only one partition of old hard drive been transferred to new MacBook

It's a pleasure to tell you that it is, possible. I've done it successfully using Winclone

You can start rebreathing ;)

Any time you experience a hard drive (not rice!) error, it's safe to assume the worst and take the initial precaution of backing up your data... Right now, before anything gets worse! Clone the whole drive and see if the clone boots.

Once you have a good backup, you can play with your old drive, attempting to reformat the entire device and see if that "fixes" the proble.

My experience leads metro suggest a real simple procedure: replace the drive before you can't get anything off it. Flakey hardware tends to fail at the worst possible time.