Is it possible for only a part of the HD of my MacBook to not work properly?
Typically, if you have a spinning hard disk drive (not an SSD or Solid State Drive) and there is a physical problem with the drive, you will get the symptoms you have described. So the first answer to your question is: yes, it is possible for only one part of a hard drive to malfunction. This has been, for me, the more frequent kind of problem, often accompanied by clicks or repetitive noises emanating from the drive itself. This may appear in the OS as one or more unreadable files (if they are in a document or user file), stuttering performance (depending on how unreadable the media is), or strange behavior by the OS (if a system file is located on a damaged portion of the disk). 

You should back up all of your data immediately. Your disk is failing and is no longer reliable. Typically, a disk with a physical continues to accumulate physical faults. Drives get more bad sectors over time, not fewer.